Matrix

Problem 1

Represent the swim meet results of Danny, John, Kenny, and Sean in a matrix format based on their placements.

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Problem 2

If $A$ and $B$ are invertible square matrices, find $(A B)^{-1}$ .

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Problem 3

Determine if a perfect hedge exists using the matrix $A$ and vector $b$ . Solve for $\hat{x}$ using $\hat{x}=(A^{} A)^{-1} A^{} b$ .

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Problem 4

Verify that $(A B)^{T} = B^{T} A^{T}$ for matrices $A=\begin{bmatrix}1 & 2 & -1 \\ 3 & 0 & 2 \\ 4 & 5 & 0\end{bmatrix}$ and $B=\begin{bmatrix}1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 1 & 3\end{bmatrix}$ .

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Problem 5

Diberikan $M=K+L^{\mathrm{T}}$ , cari nilai $y$ yang memenuhi dengan $K=\begin{pmatrix}5 & x+1 \\ 2 & y-x\end{pmatrix}$ dan $L=\begin{pmatrix}4 & 5 \\ 2x-1 & 3x+y\end{pmatrix}$ . Pilihan: a. -4, b. -3, c. -2, d. 3, e. 4.

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Problem 6

Dadas las matrices cuadradas $A$ y $B$ de orden $n$ y una constante real $k$ , determina si las siguientes afirmaciones son verdaderas.

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Problem 7

Diketahui $M=K+L^{T}$ , cari nilai $y$ dari $K=\begin{pmatrix} 5 & x+1 \\ 2 & y-x \end{pmatrix}$ dan $L=\begin{pmatrix} 4 & 5 \\ 2x-1 & 3x+y \end{pmatrix}$ . Pilihan: a. -4, b. -3, c. -2, d. 3, e. 4.

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Problem 8

Let $A$ be a $3 \times 3$ matrix. Discuss if $A$ is diagonalizable and find $T$ such that $T^{-1} A T$ is diagonal.

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Problem 9

Find eigenvalues, eigenvectors, and diagonalizing matrix $S$ for $A=\begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}$ and $B=\begin{pmatrix} 7 & 2 \\ -15 & -4 \end{pmatrix}$ .

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Problem 10

Find the inverse of matrix A using row operations: $A=\begin{bmatrix} 1 & -1 & 2 \\ 3 & 1 & 2 \\ 2 & 3 & -1 \end{bmatrix}$

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Problem 11

Explore the stress-energy tensor $T_{\alpha \beta}$ defined by
$T_{\alpha \beta}=\begin{pmatrix} \rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{pmatrix}$
with $\rho$ as energy and $p$ as pressure.

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Problem 12

Find the determinant $\Delta=\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta\end{array}\right|$ for roots $\alpha, \beta, \gamma$ of $x^{3}+ax^{2}+b=0$ .

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Problem 13

Find the inverse of the matrix $A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ .

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Problem 14

Cari nilai $x + y$ jika matriks songsang bagi $\left(\begin{array}{cc}1 & 4 \\ -3 & -7\end{array}\right)$ ialah $\frac{1}{x}\left(\begin{array}{rr}-7 & -4 \\ 3 & y\end{array}\right)$ .

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Problem 15

Calculate the product of the matrices: $\left(\begin{array}{cc}-1 & 0 \\ 6 & -5\end{array}\right)$ and $\left(\begin{array}{ll}-2 & 4 \\ -3 & 1\end{array}\right)$ .

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Problem 16

Cari nilai $x+y$ jika matriks songsang bagi $\left(\begin{array}{cc}1 & 4 \\ -3 & -7\end{array}\right)$ adalah $\frac{1}{8}\left(\begin{array}{rr}-7 & -4 \\ 3 & y\end{array}\right)$ .

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Problem 17

Cari nilai $x+y$ jika matriks songsang bagi $\left(\begin{array}{cc}1 & 4 \\ -3 & -7\end{array}\right)$ ialah $\frac{1}{8}\left(\begin{array}{rr}-7 & -4 \\ 3 & y\end{array}\right)$ .

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Problem 18

Given matrix $A = \left[\begin{array}{ccc}1 & 2 & 3 \\ 1 & -2 & -3 \\ 0 & 2 & 1\end{array}\right]$ , find $|A|$ , cofactors, adjoint, and inverse.

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Problem 19

Compute (i) $A(B+C)$ and (ii) $(B+C)A$ for matrices $A=\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}$ , $B=\begin{pmatrix}2 & 1 \\ 4 & 2\end{pmatrix}$ , $C=\begin{pmatrix}5 & 1 \\ 7 & 4\end{pmatrix}$ .

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Problem 20

Compare Company X and Company Y using provided ratios to assess risk, return, and identify the growth-oriented firm.

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Problem 21

Calculate the mean rate charged for rooms occupied that night given the rates: Double £120, Single £90, Superior £100, Double with View £140, King with View £155.

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Problem 22

Solve the system: $\begin{pmatrix} 8 & -1 \\ -7 & 2 \end{pmatrix} \begin{pmatrix} p \\ q \end{pmatrix} = \begin{pmatrix} 10 \\ -2 \end{pmatrix}$ .

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Problem 23

Solve the system of equations:
$\begin{pmatrix} -4 & 2 \\ -5 & 3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$

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Problem 24

Solve the system:
$\begin{pmatrix} -3 & 4 \\ -5 & 8 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7 \\ 12 \end{pmatrix}$

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Problem 25

A survey of 200 students on favorite sports needs table completion and analysis of popularity by gender.
a. Complete the table.
b. Identify the most and least popular sports overall.
c. Determine the most popular sport for females and males.

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Problem 26

Underline the products less than 6 from the factors $\frac{1}{10}$ , $\frac{1}{3}$ , $\frac{1}{2}$ , 1, $\frac{4}{4}$ , $\frac{3}{2}$ with product 6. What do the factors share?

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Problem 27

Choose the correct reasons for feature scaling in machine learning:
1. Speeds up gradient descent iterations.
2. Accelerates solving for $\theta$ using normal equation.
3. Prevents gradient descent from local optima.
4. Ensures matrix $X^{T} X$ is invertible.

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Problem 28

Complete the unmagic square with unique sums for rows, columns, and diagonals using digits 1-9:
2 & & 7 & 1 & 6 & & 4
Choose from options A, B, C, or D.

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Problem 29

Complete the unmagic square with different sums for rows, columns, and diagonals using digits 1-9:
8 & & 9 & 7 & 1 & & 6
Choose from options A, B, C, or D.

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Problem 30

Multiply the matrices $S=\begin{pmatrix} 4 & 1 \\ -1 & -2 \end{pmatrix}$ and $T=\begin{pmatrix} -5 & 0 \\ -4 & 3 \end{pmatrix}$ to show $S T \neq T S$ . Fill in the boxes: $S T = \_\_\_\_ \quad T S = \_\_\_\_$ .

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Problem 31

Calculate the products of matrices $S$ and $T$ : $S T$ and $T S$ to show multiplication is not commutative. Fill in the boxes.

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Problem 32

Find the product matrix $P Q$ for the matrices $P=\begin{bmatrix} 2 & 7 \\ 1 & 7 \\ 0 & 4 \\ -1 & -3 \end{bmatrix}$ and $Q=\begin{bmatrix} -2 \\ -3 \end{bmatrix}$ .

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Problem 33

Find the matrix result of $m \times H$ given the equations involving matrices and constants.

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Problem 34

Find the product matrix $B C$ for matrices $B=\begin{pmatrix}-2 & 3 \\ 0 & 5\end{pmatrix}$ and $C=\begin{pmatrix}0 & 4 & -1 \\ 3 & 0 & 1\end{pmatrix}$ .

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Problem 35

Show that matrix multiplication is not commutative by calculating $S T$ and $T S$ for the matrices $S=\begin{bmatrix} 4 & 1 \\ -3 & -2 \end{bmatrix}$ and $T=\begin{bmatrix} 0 & 4 \\ -4 & 3 \end{bmatrix}$ .

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Problem 36

Find the dimensions of the product matrix formed by multiplying any two of the following matrices: $A$ , $B$ , $C$ , $D$ .

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Problem 37

Find the matrix resulting from $m \times H$ given the equations involving matrices.

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Problem 38

Find the general solution for the system with the matrix:
$\left[\begin{array}{rrrr} 1 & 4 & 1 & 2 \\ 4 & 16 & -1 & -17 \end{array}\right]$
Choose A, B, C, or D for the solution type.

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Problem 39

Find the values of $h$ for which the matrix is consistent: $\begin{bmatrix} 1 & h & 4 \\ -3 & 6 & -16 \end{bmatrix}$ Choose A, B, C, or D.

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Problem 40

Row reduce the matrix and find pivot positions. Given matrix: $\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right]$

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Problem 41

Row reduce the matrix below to reduced echelon form and identify pivot positions.
$\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right]$

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Problem 42

Row reduce the matrix and identify pivot positions. Which option shows the correct reduced echelon form?
$\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right]$
A. $\left[\begin{array}{rrrr}1 & 0 & -1 & -2 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0\end{array}\right]$
B. $\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1\end{array}\right]$
C. $\left[\begin{array}{llll}1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 5 \\ 0 & 0 & 0 & 0\end{array}\right]$
D. $\left[\begin{array}{llll}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & 6\end{array}\right]$

See Solution

Problem 43

Find values of h for which the matrix is consistent: $\left[\begin{array}{rrr} 1 & 2 & -4 \\ 5 & h & -20 \end{array}\right]$ Choose A, B, C, or D.

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Problem 44

Identify if the following matrices are in reduced echelon form, echelon form, or neither: a. $\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 \\ 0 & 0 & 1 & 1\end{bmatrix}$ b. $\begin{bmatrix}1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0\end{bmatrix}$ c. $\begin{bmatrix}1 & 3 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$

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Problem 45

Find values of $h$ for which the matrix is consistent: $\left[\begin{array}{rrr} 1 & 4 & -3 \\ 2 & h & -6 \end{array}\right]$ Choices: A. $h \neq$ B. $h=$ C. for all $h$ D. for no $h$ .

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Problem 46

Identify if the following matrices are in reduced echelon form or just echelon form: a. $\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 1 & 1\end{bmatrix}$ b. $\begin{bmatrix}1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$ c. $\begin{bmatrix}1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$ Classify matrix a.

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Problem 47

Create a two-way table for joint and marginal relative frequencies, rounding to the nearest hundredth if needed.

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Problem 48

Complete the contingency table for 125 employees with credit card data. Fill in the missing values:
$\begin{array}{llll} & \text { Credit Card } & \text { No Credit Card } & \text { Total } \\ \text { Seasoned Employee } & & 65 & \square \\ \text { New Employee } & 9 & 89 & 125 \\ \text { Total } & 36 & \end{array}$

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Problem 49

Find the probability that a seasoned employee has a credit card, given the table: $P(\text{Credit Card | Seasoned}) = \frac{27}{92}$ . Round to two decimal places.

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Problem 50

Find the probability that a female cyclist prefers a lake path, given the table data. Round your answer to two decimal places.

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Problem 51

Decrypt the message TDGM CORF NISM NHAH WDRT NNAI ITRT ASOL E using the keyword OAK in a tabular transposition cipher.

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Problem 52

Find the values of $h$ for which the matrix is consistent:
$\left[\begin{array}{rrr} 1 & 2 & -3 \\ 5 & h & -15 \end{array}\right]$
Options: A. $h \neq$ B. $h=$ C. Consistent for all $h$ D. Not consistent for any $h$ .

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Problem 53

Identify if the matrices are in reduced echelon form or only echelon form: a. $\begin{bmatrix}1 & 1 & 0 & 1 & 1 \\ 0 & 5 & 0 & 5 & 5 \\ 0 & 0 & 0 & 3 & 3 \\ 0 & 0 & 0 & 0 & 4\end{bmatrix}$ b. $\begin{bmatrix}1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0\end{bmatrix}$ c. $\begin{bmatrix}1 & 5 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$

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Problem 54

Calculate $34 \cdot \begin{pmatrix} 2 & 5 \\ 4 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 7 \\ 3 & 0 \end{pmatrix}$ .

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Problem 55

Hitung nilai $P_{31}$ dari $P=3\begin{bmatrix}-1 & -2 \\ 2 & 4 \\ 1 & 5\end{bmatrix}$ .

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Problem 56

Subtract 3 times the first row from the matrix. Find the new first row: $\left[\begin{array}{cc}-2 & 1\end{array}\right]-3\left[\begin{array}{cc}1 & 0\end{array}\right]$

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Problem 57

Calculate the determinant of the matrix: $\Delta=\left|\begin{array}{llll}1 & 2 & 3 & 4 \\ 3 & 5 & 7 & 9 \\ 4 & 7 & 11 & 15 \\ 8 & 4 & 21 & 30\end{array}\right|$

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Problem 58

Find the product of matrices $Y=M X$ where $M=\begin{pmatrix}1 & -1 & 0 \\ -1 & 0 & 1 \\ 0 & 1 & -1\end{pmatrix}$ and $X=\begin{pmatrix}5 \\ -3 \\ 10\end{pmatrix}$ .

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Problem 59

Find the matrix $C$ given $A=\begin{bmatrix}7 & -5 \\ 10 & 6\end{bmatrix}$ and $B=\begin{bmatrix}6 & 2 \\ -18 & 14\end{bmatrix}$ , where $C=-2A-B$ .

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Problem 60

Determine the new chair of the Natural Science Division using the Borda count method for Professors D, E, F, and H.

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Problem 61

Three directors (Fred, Steve, Eddie) are voted for. Using pairwise comparison, determine who is the selected speaker.

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Problem 62

Determine the election winner using Borda count. Check if majority and head-to-head criteria are met.

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Problem 63

Given a preference table, find the winner using Borda count, and check if majority and head-to-head criteria are met.

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Problem 64

Create a preference table for candidates A, B, C. Show how the plurality-with-elimination method violates monotonicity. Votes: A=12, B=8, C=6; after eliminating C: A=14, B=12.

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Problem 65

Create a preference table for three candidates A, B, and C. Show how plurality method violates irrelevant alternatives criterion.

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Problem 66

Create the augmented matrix from $A$ and $b$ for $A \mathbf{x}=\mathbf{b}$ , then solve for $\mathbf{x}$ .

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Problem 67

Create the augmented matrix for $A \mathbf{x} = \mathbf{b}$ , then solve for $\mathbf{x}$ as a vector. Given:
$A=\begin{bmatrix} 1 & 2 & -4 \\ -3 & -3 & 3 \\ 3 & 2 & 2 \end{bmatrix}, \; b=\begin{bmatrix} 4 \\ 3 \\ -8 \end{bmatrix}$

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Problem 68

Create the augmented matrix for $A x = b$ , then solve the system and express the solution as a vector.

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Problem 69

Is the vector $u=\begin{bmatrix}-7 \\ 10 \\ 8\end{bmatrix}$ in the plane spanned by the columns of $A=\begin{bmatrix}2 & -4 \\ -5 & 7 \\ 2 & 2\end{bmatrix}$ ? Explain.

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Problem 70

Is the vector $u=\begin{bmatrix}7 \\ 4 \\ 8\end{bmatrix}$ in the plane spanned by the columns of $A=\begin{bmatrix}4 & -2 \\ -2 & 6 \\ 2 & 2\end{bmatrix}$ ? Explain.

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Problem 71

Let $A=\begin{bmatrix}2 & 1 \\ -4 & -2\end{bmatrix}$ and $\mathbf{b}=\begin{bmatrix}b_{1} \\ b_{2}\end{bmatrix}$ . Show when $A\mathbf{x}=\mathbf{b}$ has no solution and describe valid $\mathbf{b}$ .

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Problem 72

Do the columns of $A$ span $\mathbb{R}^{4}$ ? Does $A \mathbf{x}=\mathbf{b}$ have a solution for all $\mathbf{b}$ ?
$A=\begin{bmatrix} 1 & 3 & 5 & -9 \\ 2 & 0 & -8 & 6 \\ 3 & 4 & 0 & -7 \\ -2 & -7 & -13 & 18 \end{bmatrix}$

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Problem 73

Multiply matrices $A$ and $B$ and find the value of cell $\mathrm{C}_{12}$ in the product matrix $C$ .

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Problem 74

What is the value of $A B_{22}$ given the matrix multiplication result $A B = \left[\begin{array}{cc} 18 & 45 \\ 25 & 40 \end{array}\right]$ ?

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Problem 75

Find the dimension of the product of matrices sized $3 \times 4$ , $4 \times 5$ , and $5 \times 2$ .

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Problem 76

Which option best defines scalar multiplication of a matrix?
1. Some elements divided, others unchanged
2. Some elements divided, others multiplied
3. Constant divided by matrix elements
4. All elements multiplied by the constant

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Problem 77

What is the value of $A_{32}$ in the matrix $A=4\begin{pmatrix}3 & 4 & 1 \\ 5 & 2 & 6 \\ 8 & 7 & 9\end{pmatrix}$ ?

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Problem 78

Find the sum of $a_{21}+b_{21}$ from the given matrices.

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Problem 79

What is a matrix with 1s on the diagonal and 0s elsewhere called: bivariate, rotation, unity, or identity matrix?

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Problem 80

Create a 3x3 magic square using the numbers 3, 6, 9, 12, 15, 18, 21, 24, and 27.

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Problem 81

Create a 3x3 magic square using 9 of the numbers: 20, 21, 22, 23, 24, 25, 26, 27, 28, 29. Explain your solution and strategies.

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Problem 82

Check if vector $b = \begin{bmatrix} 13 \\ -7 \\ 7 \end{bmatrix}$ is a linear combo of columns of matrix $A = \begin{bmatrix} 1 & -3 & -4 \\ 0 & 6 & 3 \\ 3 & -9 & 11 \end{bmatrix}$ .

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Problem 83

Create the augmented matrix for $A \mathbf{x}=\mathbf{b}$ and solve for $\mathbf{x}$ . Given:
$A=\begin{bmatrix} 1 & 7 & -6 \\ -3 & -4 & 1 \\ -4 & -2 & 6 \end{bmatrix}, b=\begin{bmatrix} 9 \\ 7 \\ 32 \end{bmatrix}$ .

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Problem 84

Show that the equation $A x=b$ has no solution for all $b$ , and find the set of $b$ for which it does.

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Problem 85

Dataset has 154 rows (students) and 142 columns (variables). Number of rows = 154.

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Problem 86

Calculate $D(C E)$ where $C=\begin{bmatrix}3 & -7 \\ 8 & 1\end{bmatrix}$ , $D=\begin{bmatrix}9 & -2\end{bmatrix}$ , $E=\begin{bmatrix}-3 & 5 \\ -7 & 2\end{bmatrix}$ .

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Problem 87

Calculate the determinant $D$ of the matrix: $\begin{bmatrix}1 & 5 & -3 & 4 \\ 1 & 3 & -2 & 1 \\ -2 & -3 & 5 & -3 \\ 3 & 5 & -3 & -2\end{bmatrix}$ .

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Problem 88

Reduce the matrix to row echelon form:
$\begin{bmatrix} 1 & -9 & 0 & -1 & 0 & -8 \\ 0 & 1 & 0 & 0 & -2 & 3 \\ 0 & 0 & 0 & 1 & 7 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}$

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Problem 89

Is it true or false that a matrix can be row reduced to multiple reduced echelon forms? Justify your answer.

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Problem 90

Evaluate the matrix expression:
$2\begin{bmatrix} -5 & -8 \\ 6 & -7 \\ 4 & -1 \end{bmatrix} + \begin{bmatrix} -2 & -9 \\ 8 & -2 \\ 5 & 6 \end{bmatrix} - \begin{bmatrix} 5 & 5 \\ 2 & -1 \\ -9 & 0 \end{bmatrix}$
Simplify the resulting matrix.

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Problem 91

Evaluate the matrix expression:
$2\begin{pmatrix}-5 & -8 \\ 6 & -7 \\ 4 & -1\end{pmatrix} + \begin{pmatrix}-2 & -9 \\ 8 & -2 \\ 5 & 6\end{pmatrix} - \begin{pmatrix}5 & 5 \\ 2 & -1 \\ -9 & 0\end{pmatrix}$ .
Simplify the resulting matrix.

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Problem 92

Create a 3x3 magic square using the numbers 2, 5, 8, 12, 15, 18, 22, 25, and 28.

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Problem 93

Lottery machine outputs digits 0-9 in 200 trials. Find: (a) experimental probability of even numbers, (b) theoretical probability, (c) true statement about trials and probabilities. Round answers to nearest thousandths.

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Problem 94

If the price drops by \$1, how much does the total quantity demanded by Michelle, Laura, and Hillary increase? Choices: 4, 5, 2, or 3 units.

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Problem 95

Find the positive predictive value of a polygraph test: P(Lied | Positive) using the data: 13 No, 45 Yes, 30 No, 12 Yes.

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Problem 96

Calculate the correlation $r$ between Amazon and B\&N prices for 14 textbooks. Round to the nearest 0.001.

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Problem 97

A is a $3 \times 3$ matrix with three pivot positions. Does $A \mathbf{x}=\mathbf{0}$ have a nontrivial solution?

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Problem 98

Check if the matrix columns are linearly independent. Use the matrix:
$\left[\begin{array}{rrr} -4 & -3 & 0 \\ 0 & -1 & 6 \\ 1 & 1 & -6 \\ 2 & 1 & -12 \end{array}\right]$
Choose A, B, C, or D and justify your answer.

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Problem 99

Check if the matrix columns are linearly independent.
$\begin{bmatrix} 1 & 2 & -3 & 6 \\ 2 & 5 & -4 & 6 \\ 2 & 7 & 3 & -15 \end{bmatrix}$
Choose A, B, C, or D and fill in any blanks.

See Solution

Problem 100

Create an augmented matrix for the system: -3x + 7y = 8 and 5x - 8y = 4. What is the matrix's dimension?

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Matrix

Problem 1

Represent the swim meet results of Danny, John, Kenny, and Sean in a matrix format based on their placements.

Problem 2

If AAA and BBB are invertible square matrices, find (AB)−1(A B)^{-1}(AB)−1.

Problem 3

Determine if a perfect hedge exists using the matrix AAA and vector bbb. Solve for x^\hat{x}x^ using x^=(A∗A)−1A∗b\hat{x}=(A^{*} A)^{-1} A^{*} bx^=(A∗A)−1A∗b.

Problem 4

Problem 5

Diberikan M=K+LTM=K+L^{\mathrm{T}}M=K+LT, cari nilai yyy yang memenuhi dengan K=(5x+12y−x)K=\begin{pmatrix}5 & x+1 \\ 2 & y-x\end{pmatrix}K=(52​x+1y−x​) dan L=(452x−13x+y)L=\begin{pmatrix}4 & 5 \\ 2x-1 & 3x+y\end{pmatrix}L=(42x−1​53x+y​). Pilihan: a. -4, b. -3, c. -2, d. 3, e. 4.

Problem 6

Dadas las matrices cuadradas AAA y BBB de orden nnn y una constante real kkk, determina si las siguientes afirmaciones son verdaderas.

Problem 7

Diketahui M=K+LTM=K+L^{T}M=K+LT, cari nilai yyy dari K=(5x+12y−x)K=\begin{pmatrix} 5 & x+1 \\ 2 & y-x \end{pmatrix}K=(52​x+1y−x​) dan L=(452x−13x+y)L=\begin{pmatrix} 4 & 5 \\ 2x-1 & 3x+y \end{pmatrix}L=(42x−1​53x+y​). Pilihan: a. -4, b. -3, c. -2, d. 3, e. 4.

Problem 8

Let AAA be a 3×33 \times 33×3 matrix. Discuss if AAA is diagonalizable and find TTT such that T−1ATT^{-1} A TT−1AT is diagonal.

Problem 9

Find eigenvalues, eigenvectors, and diagonalizing matrix SSS for A=(1023)A=\begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}A=(12​03​) and B=(72−15−4)B=\begin{pmatrix} 7 & 2 \\ -15 & -4 \end{pmatrix}B=(7−15​2−4​).

Problem 10

Find the inverse of matrix A using row operations: A=[1−1231223−1] A=\begin{bmatrix} 1 & -1 & 2 \\ 3 & 1 & 2 \\ 2 & 3 & -1 \end{bmatrix} A=⎣⎡​132​−113​22−1​⎦⎤​

Problem 11

Problem 12

Problem 13

Find the inverse of the matrix A=[111011001]A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}A=⎣⎡​100​110​111​⎦⎤​.

Problem 14

Cari nilai x+yx + yx+y jika matriks songsang bagi (14−3−7)\left(\begin{array}{cc}1 & 4 \\ -3 & -7\end{array}\right)(1−3​4−7​) ialah 1x(−7−43y)\frac{1}{x}\left(\begin{array}{rr}-7 & -4 \\ 3 & y\end{array}\right)x1​(−73​−4y​).

Problem 15

Calculate the product of the matrices: (−106−5)\left(\begin{array}{cc}-1 & 0 \\ 6 & -5\end{array}\right)(−16​0−5​) and (−24−31)\left(\begin{array}{ll}-2 & 4 \\ -3 & 1\end{array}\right)(−2−3​41​).

Problem 16

Cari nilai x+yx+yx+y jika matriks songsang bagi (14−3−7)\left(\begin{array}{cc}1 & 4 \\ -3 & -7\end{array}\right)(1−3​4−7​) adalah 18(−7−43y)\frac{1}{8}\left(\begin{array}{rr}-7 & -4 \\ 3 & y\end{array}\right)81​(−73​−4y​).

Problem 17

Cari nilai x+yx+yx+y jika matriks songsang bagi (14−3−7)\left(\begin{array}{cc}1 & 4 \\ -3 & -7\end{array}\right)(1−3​4−7​) ialah 18(−7−43y)\frac{1}{8}\left(\begin{array}{rr}-7 & -4 \\ 3 & y\end{array}\right)81​(−73​−4y​).

Problem 18

Given matrix A=[1231−2−3021]A = \left[\begin{array}{ccc}1 & 2 & 3 \\ 1 & -2 & -3 \\ 0 & 2 & 1\end{array}\right]A=⎣⎡​110​2−22​3−31​⎦⎤​, find ∣A∣|A|∣A∣, cofactors, adjoint, and inverse.

Problem 19

Compute (i) A(B+C)A(B+C)A(B+C) and (ii) (B+C)A(B+C)A(B+C)A for matrices A=(1234)A=\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}A=(13​24​), B=(2142)B=\begin{pmatrix}2 & 1 \\ 4 & 2\end{pmatrix}B=(24​12​), C=(5174)C=\begin{pmatrix}5 & 1 \\ 7 & 4\end{pmatrix}C=(57​14​).

Problem 20

Compare Company X and Company Y using provided ratios to assess risk, return, and identify the growth-oriented firm.

Problem 21

Calculate the mean rate charged for rooms occupied that night given the rates: Double £120, Single £90, Superior £100, Double with View £140, King with View £155.

Problem 22

Solve the system: (8−1−72)(pq)=(10−2)\begin{pmatrix} 8 & -1 \\ -7 & 2 \end{pmatrix} \begin{pmatrix} p \\ q \end{pmatrix} = \begin{pmatrix} 10 \\ -2 \end{pmatrix}(8−7​−12​)(pq​)=(10−2​).

Problem 23

Solve the system of equations: (−42−53)(xy)=(12) \begin{pmatrix} -4 & 2 \\ -5 & 3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} (−4−5​23​)(xy​)=(12​)

Problem 24

Solve the system: (−34−58)(xy)=(712) \begin{pmatrix} -3 & 4 \\ -5 & 8 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7 \\ 12 \end{pmatrix} (−3−5​48​)(xy​)=(712​)

Problem 25

A survey of 200 students on favorite sports needs table completion and analysis of popularity by gender. a. Complete the table.b. Identify the most and least popular sports overall.c. Determine the most popular sport for females and males.

Problem 26

Underline the products less than 6 from the factors 110\frac{1}{10}101​, 13\frac{1}{3}31​, 12\frac{1}{2}21​, 1, 44\frac{4}{4}44​, 32\frac{3}{2}23​ with product 6. What do the factors share?

Problem 27

Choose the correct reasons for feature scaling in machine learning:1. Speeds up gradient descent iterations.2. Accelerates solving for θ\thetaθ using normal equation.3. Prevents gradient descent from local optima.4. Ensures matrix XTXX^{T} XXTX is invertible.

Problem 28

Complete the unmagic square with unique sums for rows, columns, and diagonals using digits 1-9: 2 & & 7 & 1 & 6 & & 4 Choose from options A, B, C, or D.

Problem 29

Complete the unmagic square with different sums for rows, columns, and diagonals using digits 1-9: 8 & & 9 & 7 & 1 & & 6 Choose from options A, B, C, or D.

Problem 30

Problem 31

Calculate the products of matrices SSS and TTT: STS TST and TST STS to show multiplication is not commutative. Fill in the boxes.

Problem 32

Find the product matrix PQP QPQ for the matrices P=[271704−1−3]P=\begin{bmatrix} 2 & 7 \\ 1 & 7 \\ 0 & 4 \\ -1 & -3 \end{bmatrix}P=⎣⎡​210−1​774−3​⎦⎤​ and Q=[−2−3]Q=\begin{bmatrix} -2 \\ -3 \end{bmatrix}Q=[−2−3​].

Problem 33

Find the matrix result of m×Hm \times Hm×H given the equations involving matrices and constants.

Problem 34

Find the product matrix BCB CBC for matrices B=(−2305)B=\begin{pmatrix}-2 & 3 \\ 0 & 5\end{pmatrix}B=(−20​35​) and C=(04−1301)C=\begin{pmatrix}0 & 4 & -1 \\ 3 & 0 & 1\end{pmatrix}C=(03​40​−11​).

Problem 35

Show that matrix multiplication is not commutative by calculating STS TST and TST STS for the matrices S=[41−3−2]S=\begin{bmatrix} 4 & 1 \\ -3 & -2 \end{bmatrix}S=[4−3​1−2​] and T=[04−43]T=\begin{bmatrix} 0 & 4 \\ -4 & 3 \end{bmatrix}T=[0−4​43​].

Problem 36

Find the dimensions of the product matrix formed by multiplying any two of the following matrices: AAA, BBB, CCC, DDD.

Problem 37

Find the matrix resulting from m×Hm \times Hm×H given the equations involving matrices.

Problem 38

Find the general solution for the system with the matrix: [1412416−1−17] \left[\begin{array}{rrrr} 1 & 4 & 1 & 2 \\ 4 & 16 & -1 & -17 \end{array}\right] [14​416​1−1​2−17​] Choose A, B, C, or D for the solution type.

Problem 39

Find the values of hhh for which the matrix is consistent: [1h4−36−16] \begin{bmatrix} 1 & h & 4 \\ -3 & 6 & -16 \end{bmatrix} [1−3​h6​4−16​] Choose A, B, C, or D.

Problem 40

Row reduce the matrix and find pivot positions. Given matrix: [123456786789] \left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right] ⎣⎡​156​267​378​489​⎦⎤​

Problem 41

Row reduce the matrix below to reduced echelon form and identify pivot positions.[123456786789] \left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right] ⎣⎡​156​267​378​489​⎦⎤​

Problem 42

If $A$ and $B$ are invertible square matrices, find $(A B)^{-1}$ .

Determine if a perfect hedge exists using the matrix $A$ and vector $b$ . Solve for $\hat{x}$ using $\hat{x}=(A^{} A)^{-1} A^{} b$ .

Diberikan $M=K+L^{\mathrm{T}}$ , cari nilai $y$ yang memenuhi dengan $K=\begin{pmatrix}5 & x+1 \\ 2 & y-x\end{pmatrix}$ dan $L=\begin{pmatrix}4 & 5 \\ 2x-1 & 3x+y\end{pmatrix}$ . Pilihan: a. -4, b. -3, c. -2, d. 3, e. 4.

Dadas las matrices cuadradas $A$ y $B$ de orden $n$ y una constante real $k$ , determina si las siguientes afirmaciones son verdaderas.

Diketahui $M=K+L^{T}$ , cari nilai $y$ dari $K=\begin{pmatrix} 5 & x+1 \\ 2 & y-x \end{pmatrix}$ dan $L=\begin{pmatrix} 4 & 5 \\ 2x-1 & 3x+y \end{pmatrix}$ . Pilihan: a. -4, b. -3, c. -2, d. 3, e. 4.

Let $A$ be a $3 \times 3$ matrix. Discuss if $A$ is diagonalizable and find $T$ such that $T^{-1} A T$ is diagonal.

Find eigenvalues, eigenvectors, and diagonalizing matrix $S$ for $A=\begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}$ and $B=\begin{pmatrix} 7 & 2 \\ -15 & -4 \end{pmatrix}$ .

Find the inverse of matrix A using row operations: $A=\begin{bmatrix} 1 & -1 & 2 \\ 3 & 1 & 2 \\ 2 & 3 & -1 \end{bmatrix}$

Find the inverse of the matrix $A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ .

Cari nilai $x + y$ jika matriks songsang bagi $\left(\begin{array}{cc}1 & 4 \\ -3 & -7\end{array}\right)$ ialah $\frac{1}{x}\left(\begin{array}{rr}-7 & -4 \\ 3 & y\end{array}\right)$ .

Calculate the product of the matrices: $\left(\begin{array}{cc}-1 & 0 \\ 6 & -5\end{array}\right)$ and $\left(\begin{array}{ll}-2 & 4 \\ -3 & 1\end{array}\right)$ .

Cari nilai $x+y$ jika matriks songsang bagi $\left(\begin{array}{cc}1 & 4 \\ -3 & -7\end{array}\right)$ adalah $\frac{1}{8}\left(\begin{array}{rr}-7 & -4 \\ 3 & y\end{array}\right)$ .

Cari nilai $x+y$ jika matriks songsang bagi $\left(\begin{array}{cc}1 & 4 \\ -3 & -7\end{array}\right)$ ialah $\frac{1}{8}\left(\begin{array}{rr}-7 & -4 \\ 3 & y\end{array}\right)$ .

Given matrix $A = \left[\begin{array}{ccc}1 & 2 & 3 \\ 1 & -2 & -3 \\ 0 & 2 & 1\end{array}\right]$ , find $|A|$ , cofactors, adjoint, and inverse.

Compute (i) $A(B+C)$ and (ii) $(B+C)A$ for matrices $A=\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}$ , $B=\begin{pmatrix}2 & 1 \\ 4 & 2\end{pmatrix}$ , $C=\begin{pmatrix}5 & 1 \\ 7 & 4\end{pmatrix}$ .

Solve the system: $\begin{pmatrix} 8 & -1 \\ -7 & 2 \end{pmatrix} \begin{pmatrix} p \\ q \end{pmatrix} = \begin{pmatrix} 10 \\ -2 \end{pmatrix}$ .

Solve the system of equations:
$\begin{pmatrix} -4 & 2 \\ -5 & 3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$

Solve the system:
$\begin{pmatrix} -3 & 4 \\ -5 & 8 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7 \\ 12 \end{pmatrix}$

A survey of 200 students on favorite sports needs table completion and analysis of popularity by gender.
a. Complete the table.
b. Identify the most and least popular sports overall.
c. Determine the most popular sport for females and males.

Underline the products less than 6 from the factors $\frac{1}{10}$ , $\frac{1}{3}$ , $\frac{1}{2}$ , 1, $\frac{4}{4}$ , $\frac{3}{2}$ with product 6. What do the factors share?

Choose the correct reasons for feature scaling in machine learning:
1. Speeds up gradient descent iterations.
2. Accelerates solving for $\theta$ using normal equation.
3. Prevents gradient descent from local optima.
4. Ensures matrix $X^{T} X$ is invertible.

Complete the unmagic square with unique sums for rows, columns, and diagonals using digits 1-9:
2 & & 7 & 1 & 6 & & 4
Choose from options A, B, C, or D.

Complete the unmagic square with different sums for rows, columns, and diagonals using digits 1-9:
8 & & 9 & 7 & 1 & & 6
Choose from options A, B, C, or D.

Calculate the products of matrices $S$ and $T$ : $S T$ and $T S$ to show multiplication is not commutative. Fill in the boxes.

Find the product matrix $P Q$ for the matrices $P=\begin{bmatrix} 2 & 7 \\ 1 & 7 \\ 0 & 4 \\ -1 & -3 \end{bmatrix}$ and $Q=\begin{bmatrix} -2 \\ -3 \end{bmatrix}$ .

Find the matrix result of $m \times H$ given the equations involving matrices and constants.

Find the product matrix $B C$ for matrices $B=\begin{pmatrix}-2 & 3 \\ 0 & 5\end{pmatrix}$ and $C=\begin{pmatrix}0 & 4 & -1 \\ 3 & 0 & 1\end{pmatrix}$ .

Show that matrix multiplication is not commutative by calculating $S T$ and $T S$ for the matrices $S=\begin{bmatrix} 4 & 1 \\ -3 & -2 \end{bmatrix}$ and $T=\begin{bmatrix} 0 & 4 \\ -4 & 3 \end{bmatrix}$ .

Find the dimensions of the product matrix formed by multiplying any two of the following matrices: $A$ , $B$ , $C$ , $D$ .

Find the matrix resulting from $m \times H$ given the equations involving matrices.

Find the general solution for the system with the matrix:
$\left[\begin{array}{rrrr} 1 & 4 & 1 & 2 \\ 4 & 16 & -1 & -17 \end{array}\right]$
Choose A, B, C, or D for the solution type.

Find the values of $h$ for which the matrix is consistent: $\begin{bmatrix} 1 & h & 4 \\ -3 & 6 & -16 \end{bmatrix}$ Choose A, B, C, or D.

Row reduce the matrix and find pivot positions. Given matrix: $\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right]$

Row reduce the matrix below to reduced echelon form and identify pivot positions.
$\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 6 & 7 & 8 & 9 \end{array}\right]$

Find values of h for which the matrix is consistent: $\left[\begin{array}{rrr} 1 & 2 & -4 \\ 5 & h & -20 \end{array}\right]$ Choose A, B, C, or D.

Find values of $h$ for which the matrix is consistent: $\left[\begin{array}{rrr} 1 & 4 & -3 \\ 2 & h & -6 \end{array}\right]$ Choices: A. $h \neq$ B. $h=$ C. for all $h$ D. for no $h$ .

Find the probability that a seasoned employee has a credit card, given the table: $P(\text{Credit Card | Seasoned}) = \frac{27}{92}$ . Round to two decimal places.

Find the values of $h$ for which the matrix is consistent:
$\left[\begin{array}{rrr} 1 & 2 & -3 \\ 5 & h & -15 \end{array}\right]$
Options: A. $h \neq$ B. $h=$ C. Consistent for all $h$ D. Not consistent for any $h$ .

Calculate $34 \cdot \begin{pmatrix} 2 & 5 \\ 4 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 7 \\ 3 & 0 \end{pmatrix}$ .

Hitung nilai $P_{31}$ dari $P=3\begin{bmatrix}-1 & -2 \\ 2 & 4 \\ 1 & 5\end{bmatrix}$ .

Subtract 3 times the first row from the matrix. Find the new first row: $\left[\begin{array}{cc}-2 & 1\end{array}\right]-3\left[\begin{array}{cc}1 & 0\end{array}\right]$

Calculate the determinant of the matrix: $\Delta=\left|\begin{array}{llll}1 & 2 & 3 & 4 \\ 3 & 5 & 7 & 9 \\ 4 & 7 & 11 & 15 \\ 8 & 4 & 21 & 30\end{array}\right|$

Find the product of matrices $Y=M X$ where $M=\begin{pmatrix}1 & -1 & 0 \\ -1 & 0 & 1 \\ 0 & 1 & -1\end{pmatrix}$ and $X=\begin{pmatrix}5 \\ -3 \\ 10\end{pmatrix}$ .

Find the matrix $C$ given $A=\begin{bmatrix}7 & -5 \\ 10 & 6\end{bmatrix}$ and $B=\begin{bmatrix}6 & 2 \\ -18 & 14\end{bmatrix}$ , where $C=-2A-B$ .

Create the augmented matrix from $A$ and $b$ for $A \mathbf{x}=\mathbf{b}$ , then solve for $\mathbf{x}$ .

Create the augmented matrix for $A x = b$ , then solve the system and express the solution as a vector.

Is the vector $u=\begin{bmatrix}-7 \\ 10 \\ 8\end{bmatrix}$ in the plane spanned by the columns of $A=\begin{bmatrix}2 & -4 \\ -5 & 7 \\ 2 & 2\end{bmatrix}$ ? Explain.

Is the vector $u=\begin{bmatrix}7 \\ 4 \\ 8\end{bmatrix}$ in the plane spanned by the columns of $A=\begin{bmatrix}4 & -2 \\ -2 & 6 \\ 2 & 2\end{bmatrix}$ ? Explain.

Let $A=\begin{bmatrix}2 & 1 \\ -4 & -2\end{bmatrix}$ and $\mathbf{b}=\begin{bmatrix}b_{1} \\ b_{2}\end{bmatrix}$ . Show when $A\mathbf{x}=\mathbf{b}$ has no solution and describe valid $\mathbf{b}$ .

Multiply matrices $A$ and $B$ and find the value of cell $\mathrm{C}_{12}$ in the product matrix $C$ .

What is the value of $A B_{22}$ given the matrix multiplication result $A B = \left[\begin{array}{cc} 18 & 45 \\ 25 & 40 \end{array}\right]$ ?