Linear Algebra

Problem 1

Solve the matrix equation $\left(\begin{array}{ll}6 & 1 \\ 4 & 3\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}-3 \\ 5\end{array}\right)$ for $x$ and $y$ .

See Solution

Problem 2

Determine if the given $2 \times 4$ matrix is in row reduced form.

See Solution

Problem 3

Write an inequality to represent the minimum cargo weight requirement, where $x$ is the number of shorter containers (45,600 lbs each) and $y$ is the number of longer containers (56,000 lbs each), with a total cargo weight of at least 785,000 lbs.

See Solution

Problem 4

Solve the system using the given inverse coefficient matrix: $x + 2y + 3z = -3$ , $x + y + z = 4$ , $-x + y + 2z = -2$ .

See Solution

Problem 5

Write a system of linear inequalities with two lines: a solid line starting at $y=1$ with slope $-5/1$ , and a dashed line starting at $y=-2$ with slope $1$ . Represent the shaded region between the lines.

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Problem 6

Determine if the solution region for the given system of linear inequalities is bounded or unbounded.
$-x + 2y \geq 4, 2x - y \leq 2, x \geq 0, y \geq 0$

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Problem 7

Find the image coordinates of a point $Q(-5,1)$ under the transformation $(x, y) \rightarrow(-y,-x)$ .

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Problem 8

Fill in the missing values in the matrix equation $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} ? \\ ? \\ ? \end{bmatrix}$ .

See Solution

Problem 9

Find the value of $k$ that makes the vectors $(2,-6,4)$ and $(1, k,-2)$ collinear.

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Problem 10

Determine if the set of all symmetric $2 \times 2$ matrices, $W$ , is a subspace of $R^{2 \times 2}$ .

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Problem 11

Find the eigenvalues of the $2 \times 2$ matrix $A = \left[\begin{array}{cc}3 & 0 \\ -1 & 8\end{array}\right]$ .

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Problem 12

If a set of 3 vectors $\{v_1, v_2, v_3\}$ spans $\mathbb{R}^3$ , then it forms a basis for $\mathbb{R}^3$ . True or False?

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Problem 13

Find the plane equation $ax + by + cz = 0$ that passes through the points $(1,0,2), (-1,1,-2)$ , and the origin.

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Problem 14

Find the value of $c$ if the vectors $v_1 = [1, 0, 0]$ , $v_2 = [1, 1, 0]$ , and $v_3 = [1, -1, c]$ are linearly dependent.

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Problem 15

Given constraints and optimal solution $(x, y) = (25, 0)$ , find the values of the slack variables $s_1$ and $s_2$ .

See Solution

Problem 16

Convert 1973.1 cubic miles to cubic kilometers using the conversion ratio: $1.61 \mathrm{~km} = 1$ mile.

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Problem 17

Solve for matrix $X$ in $(A-5B)X=C$ given $A=\left[\begin{array}{cc}-5 & 0 \\ 1 & 2\end{array}\right], B=\left[\begin{array}{cc}-1 & 2 \\ 0 & 3\end{array}\right], C=\left[\begin{array}{cc}-2 & -6 \\ -3 & -5\end{array}\right]$ .

See Solution

Problem 18

Finden Sie einen Vektor $\vec{n}$ , der senkrecht auf dem Vektor $\vec{a}=\left(\begin{array}{c}2 \\ 5 \\ -3\end{array}\right)$ steht.

See Solution

Problem 19

Find the vector $-6(-30\mathbf{i}+40\mathbf{j})$ .

See Solution

Problem 20

Find the y-value of the solution to the $2 \times 2$ matrix equation $\left[\begin{array}{ll} 3 & 2 \\ 2 & 1 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{l} 3 \\ 4 \end{array}\right]$ .

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Problem 21

Solve for $\mathbf{x}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ using $\mathbf{x}+6\mathbf{a}-\mathbf{b}=6(\mathbf{x}+\mathbf{a})-(2\mathbf{a}-\mathbf{b})$ .

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Problem 22

Is the set $W=\{(x_1, 0, x_3): x_1, x_3 \text{ are real numbers}\}$ a subspace of $\mathbb{R}^3$ ?

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Problem 23

If a square matrix is symmetric, its transpose is also symmetric. True or False?

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Problem 24

Find the rank of $2 \times 2$ matrix $A$ if $Ax=0$ has only the zero solution.

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Problem 25

Solve linear system $Ax=b$ where $A$ is $3 \times 3$ matrix, $x=\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right]$ , $b=\left[\begin{array}{l}-2 \\ -4 \\ -3\end{array}\right]$ , and $A^{-1}=\left[\begin{array}{ccc}1 & 4 & -1 \\ 2 & -3 & 2 \\ 0 & 5 & -4\end{array}\right]$ . Find the value of $x_{3}$ .

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Problem 26

Find the value of $b$ where the linear system with augmented matrix $\left[\begin{array}{lllll}1 & 0 & 0 & \mid & 1 \\ 0 & 1 & b & \mid & 2 \\ 0 & 2 & 4 & \mid & 4\end{array}\right]$ has infinitely many solutions.

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Problem 27

Find the values of $s$ , $t$ , and $r$ in the given $2\times 2$ matrix equation.

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Problem 28

Determine if the lines $y=\frac{-5}{6}x+\frac{9}{4}$ and $y=\frac{6}{5}x+\frac{5}{2}$ are parallel, perpendicular, or neither.

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Problem 29

Is it possible for the consistent system $Ax=b$ to have multiple solutions for some $b$ ? Yes, sometimes. No, never.

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Problem 30

Convert augmented matrix to system of 3 linear equations in $x$ , $y$ , and $z$ with constant terms.

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Problem 31

Find values of $r$ and $s$ that satisfy $3[2r, 5s] - [s, 3r] = [-18, 9]$ . If no solution exists, state "Not Possible".

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Problem 32

Find $C-A+B$ and simplify, if possible. Give exact answers in fraction form, if necessary. Select "Undefined" if applicable.

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Problem 33

Choose a system of linear equations to represent the growth of a $4$ inch per year spruce tree and a $6$ inch per year hemlock tree with initial heights of $14$ inches and $8$ inches, respectively.

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Problem 34

Solve the system of linear equations represented by the given augmented matrix using Gauss-Jordan method. Perform the row operations and find the values of $x$ , $y$ , and $z$ .

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Problem 35

Find the transition matrix from basis $F$ to basis $E$ where $E=[(1,2)^{T},(3,7)^{T}]$ and $F=[(1,-2)^{T},(3,-5)^{T}]$ in $\mathbb{R}^{2}$ .

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Problem 36

Solve the linear system $Ax=b$ where $A$ is a $3\times 3$ matrix, $x=\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}$ , and $b=\begin{bmatrix}5\\0\\-5\end{bmatrix}$ . If $A^{-1}=\begin{bmatrix}1&4&-1\\2&-3&2\\0&5&-4\end{bmatrix}$ , then $x_3=$

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Problem 37

Find $z$ using Cramer's rule and a calculator for the system: $4x-y+5z=1, -x+2y+z=0, 5x-2y+2z=-4$ .

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Problem 38

Solve the system $A \mathbf{x}=\mathbf{b}$ using the given factorization $A=P^{\top} L U$ , where $\mathbf{b}=\begin{bmatrix}1 \\ 1 \\ 7\end{bmatrix}$ and $\mathbf{x}=\begin{bmatrix}2 \\ 3 \\ 2\end{bmatrix}$ .

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Problem 39

Find the matrix $X$ that satisfies the equation $A(X + 5B) = C$ .

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Problem 40

Find the inverse $A^{-1}$ of the given 3x3 matrix $A = \begin{bmatrix} 1 & 1 & 3 \\ -2 & -1 & -6 \\ 0 & 3 & 1 \end{bmatrix}$ .

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Problem 41

Solve the given systems of linear equations (a) $4x_1 - 3x_2 + 6x_3 = 0, 2x_1 - x_3 = 5, 4x_1 = -2$ , (b) $4x_1 - x_2 + 3x_3 = 2, x_1 + 3x_2 = 5, 4x_2 = 8$ , (c) $5x_1 = 10, 5x_2 - 3x_3 = 9, 4x_1 + x_2 = 0$ , (d) $a + b = 3, a + b - c = 0, b + c = 4$ , (e) $r + s - t = 0, r + t = 2, r - 2s + t = 2$ , (f) $0.6y + 1.8z = 3, 0.3x + 1.2y = 0, 0.5x + z = 1$ .

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Problem 42

Find the most accurate statement about the given $6 \times 6$ matrix $A$ , which could be the adjacency matrix of a graph or a digraph.

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Problem 43

Given a vector from (4,1) to (5,5), express the vector in component form. Assume each grid box is 1x1 unit.

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Problem 44

Maximize $z=3x_1+7x_2+8x_3$ subject to $5x_1-x_2+x_3\leq1500, 2x_1+2x_2+x_3\leq2500, 4x_1+2x_2+x_3\leq2000, x_1,x_2,x_3\geq0$ .

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Problem 45

Find the intersection point of 3 one-way streets using Gaussian elimination to solve the system of $x+9=y+13$ , $z+15=x+10$ , $y+23=z+24$ .

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Problem 46

Find a vector $\mathbf{v}$ with magnitude 4, where the $\mathbf{i}$ component is twice the $\mathbf{j}$ component.

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Problem 47

Determine if $3 \times 3$ matrices $A = \left[\begin{array}{rr} 3 & -1 \\ -20 & 10 \end{array}\right]$ and $B = \left[\begin{array}{cc} 1 & \frac{1}{10} \\ 2 & \frac{3}{10} \end{array}\right]$ are inverses by computing their product.

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Problem 48

Solve for matrix $A$ in the equation $A\left[\begin{array}{lll} 0 & 6 & 6 \\ 1 & 2 & 2 \\ 0 & 7 & 8 \end{array}\right]=\left[\begin{array}{ccc} -9 & 1 & 0 \\ -2 & -2 & 8 \\ 6 & 5 & -6 \end{array}\right]$ , where $A=\left[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$ .

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Problem 49

Given the $3 \times 4$ matrix $D$ , find the value of the element $d_{23}$ . If the element is undefined, press the undefined button.

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Problem 50

Find the value(s) of $k$ that satisfy the matrix equation $\left[\begin{array}{ccc}2 & 2 & k\end{array}\right]\left[\begin{array}{ccc}1 & 3 & 0\\3 & 0 & 5\\0 & 5 & -1\end{array}\right]\left[\begin{array}{l}2\\2\\k\end{array}\right]=0$ .

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Problem 51

Find the least squares line for the set of points $(x, y)$ where $x \in [1, 10]$ and $y = [0.4, 0.9, 1.4, 2.4, 2.9, 3.4, 3.4, 4.4, 4.9, 5.4]$ .

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Problem 52

Finde die Funktionsgleichung einer linearen Funktion mit Steigung $m=-2$ , die durch den Punkt $\mathbf{P (-7/-5)}$ verläuft.

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Problem 53

Solve the system of 3 linear equations in 3 variables: $-3x+4y+2z=-28, x+5y-4z=-6, 4x+y-6z=22$ .

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Problem 54

Perform matrix operations on $A$ , $B$ , $C$ , and $D$ matrices, such as $B+D$ and $B-A$ .

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Problem 55

Compute the product of the given $3 \times 1$ and $1 \times 3$ matrices.

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Problem 56

Find the value of the $3 \times 3$ determinant: $\left|\begin{array}{ccc} 3 & -3 & 0 \\ 1 & -2 & 1 \\ -2 & 1 & 2 \end{array}\right|$

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Problem 57

Solve the system of linear equations using Gaussian elimination: $2x + y - z = 2, x + 3y + 2z = 1, x + y + z = 2$ .

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Problem 58

Which linear transformation from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$ is defined by $T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]\right)=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]$ ?

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Problem 59

Find the determinant of the matrix product of $A=\left[\begin{array}{ll}4 & 2\end{array}\right]$ and $B=\left[\begin{array}{c}-1 \\ 3\end{array}\right]$ .

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Problem 60

Find the value(s) of $a$ such that the $3\\times 3$ matrix $A=\left[\begin{array}{lll}2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 1 & a\end{array}\right]$ has no inverse.

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Problem 61

Find the matrix product of -3 and a 2x2 matrix $E=\begin{bmatrix}-4 & 5 \\ 6 & -1\end{bmatrix}$ .

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Problem 62

Find the basis and dimension of the subspace $V \subset \mathbb{R}^{4}$ given by the homogeneous system $x_1 + 2x_2 + x_3 + 2x_4 = 0, 3x_1 + 5x_2 + 4x_3 + 7x_4 = 0$ . Then, find the homogeneous system of linear equations that define the subspace $W \subset \mathbb{R}^{4}$ spanned by the basis of $V$ and the vector $w = (1, 0, 0, 0)$ .

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Problem 63

Find the product of matrix $C$ with $x$ and matrix $D$ with $y$ , then add the results.

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Problem 64

Find the min and max of $z=3x+9y$ subject to $3x+4y \geq 12$ , $x+4y \geq 8$ , $x \geq 0$ , $y \geq 0$ . (Round min to nearest tenth.)

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Problem 65

Find the product of two $2 \times 2$ matrices $X$ and $Y$ , where $X=\left[\begin{array}{cc}4 & 4 \\ -3 & -1\end{array}\right]$ and $Y=\left[\begin{array}{cc}4 & 1 \\ 4 & -1\end{array}\right]$ .

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Problem 66

Find vectors parallel to $\left(\begin{array}{c}-2 \\ 4\end{array}\right)$ .

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Problem 67

Maximize linear function $p = x - 2y$ subject to linear constraints. Determine if function is unbounded.

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Problem 68

Use Cramer's rule to find the value of $y$ that satisfies the given system of 3 linear equations in 3 variables.

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Problem 69

Solve the linear programming problem using the simplex method to maximize $P=x+4y-2z$ subject to $3x+y-z\leq44$ , $2x+y-z\leq22$ , $-x+y+z\leq44$ , and $x,y,z\geq0$ .

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Problem 70

Write a system of linear equations from the given augmented matrix, then solve using back-substitution. Variables: $x, y, z$ .

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Problem 71

Perform the given row operations on the provided $2 \times 3$ matrix.

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Problem 72

Find $\frac{1}{3} B+2 A$ given $A=\left[\begin{array}{ll}6 & 3\end{array}\right]$ and $B=\left[\begin{array}{ll}-15 & 6\end{array}\right]$ .

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Problem 73

Find the difference between two 2x2 matrices $A$ and $B$ .

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Problem 74

Find the product of two $2 \times 2$ matrices: $\left[\begin{array}{cc}-2 & 0\\0 & -4\\0 & 3\\-2 & 0\end{array}\right]$ and $\left[\begin{array}{cc}-1 & -1\\3 & -1\end{array}\right]$ .

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Problem 75

Rewrite rotation formulas to eliminate $x'y'$ -term in equation $x^2 + 16xy + y^2 - 3 = 0$ .

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Problem 76

Find the value of the objective function $z=5x+8y$ at the corner points of the region where $x\geq 0, y\geq 0, 2x+y\leq 12, x+y\geq 6$ .

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Problem 77

Determine the value of $d_{23}$ in the given $2 \times 3$ matrix $D$ . If $d_{23}$ is not present, state "None".

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Problem 78

Find the vector $4\mathbf{v}$ given $\mathbf{v}=\langle-6,0\rangle$ .

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Problem 79

Find the dimension and value of the matrix element $a_{12}$ .

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Problem 80

Find the unit vector in the direction of the vector $\langle-6,5\rangle$ .

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Problem 81

Find the closest vector $\mathbf{x}$ in the subspace $W$ spanned by $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ to the given vector $\mathbf{y}$ , where $\mathbf{y}=\left[\begin{array}{r}1 \\ -1 \\ 1 \\ 19\end{array}\right]$ , $\mathbf{v}_{1}=\left[\begin{array}{r}1 \\ -1 \\ -1 \\ 1\end{array}\right]$ , and $\mathbf{v}_{2}=\left[\begin{array}{r}-1 \\ 1 \\ 0 \\ 2\end{array}\right]$ .

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Problem 82

Find a symmetric matrix $A$ with non-negative eigenvalues and show that $A$ has a symmetric square root $R$ such that $R^2 = A$ . Then, find two different square roots of the matrix $B = \left[\begin{array}{cc}5 & -3 \\ -3 & 5\end{array}\right]$ .

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Problem 83

Find the dot product of the vectors $\mathbf{u} = 2\mathbf{i} - 3\mathbf{j}$ and $\mathbf{v} = 5\mathbf{j}$ .

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Problem 84

Solve the matrix equation for $Y$ . Simplify all elements. Find the size of $Y$ if the matrix exists, or select undefined.

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Problem 85

Minimize $c=x+2y$ subject to $x+5y \geq 19, 4x+y \geq 19, x \geq 0, y \geq 0$ , where $c = (x,y)=(\sqrt{x})$ .

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Problem 86

Compute the product of two 3x3 matrices: $\left[\begin{array}{ccc}0 & 0 & 4 \\ -2 & -2 & 2 \\ 0 & -2 & -5\end{array}\right]$ and $\left[\begin{array}{ccc}-1 & -1 & -1 \\ 1 & -3 & 0 \\ 0 & 1 & 1\end{array}\right]$ .

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Problem 87

Solve the system of linear equations and evaluate the determinants $D$ , $D_x$ , and $D_y$ .
$6(x+y) = 9x + 6 \\ 6x = 12y - 12$
$D=\square \quad D_x=\square \quad D_y=\square$

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Problem 88

Solve the system of linear equations $x + y + 2z = 1$ , $x - y = 1$ , and $x - z = 2$ .

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Problem 89

Find the maximum value of $z=4x+5y$ subject to the constraints $3x+4y\leq40$ , $7x+3y\leq49$ , and $y\geq0$ using the Desmos Graphing Calculator 2.

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Problem 90

Find the determinant of the $3 \times 3$ matrix with entries $-1, 1, 0; 1, 0, 2; 1, -3, -1$ .

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Problem 91

Solve the initial tableau of a linear programming problem using the simplex method. The maximum is $-3$ when $x_1=\frac{11}{5}$ , $x_2=0$ , $x_3=\frac{11}{4}$ , $s_1=\frac{11}{2}$ , and $s_2=0$ .

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Problem 92

Zadanie 5. Given $\mathcal{A}=\{(1,4),(1,5)\}, \mathcal{B}=\{(2,1),(5,2)\}$ and linear transformation $\varphi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ with $M(\varphi)_{\mathcal{A}}^{\mathcal{B}}=\left[\begin{array}{ll}1 & 0 \\ 2 & 3\end{array}\right]$ , find $M(\varphi \circ \varphi)_{\mathcal{A}}^{\mathcal{B}}$ .

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Problem 93

Find the range of the set $A = \{(2,1), (2,3), (-1,4), (1,1)\}$ .

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Problem 94

Find the values of $a$ for which the $2 \times 2$ matrix $\begin{bmatrix} 2-a & 4 \\ 3 & 3-a \end{bmatrix}$ is singular.

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Problem 95

Find the absolute value of $3A$ if $A$ is a $3 \times 3$ matrix and $\frac{8}{9} = |\!2A^T\!|$ .

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Problem 96

Find the least squares solution to $Ax=b$ where $A=\begin{bmatrix}3&-5\\1&-7\\0&1\end{bmatrix}$ and $b=\begin{bmatrix}0\\2\\5\end{bmatrix}$ . Options: a. $\begin{bmatrix}-\frac{24}{266}\\\frac{23}{266}\end{bmatrix}$ b. $\begin{bmatrix}-24\\-23\end{bmatrix}$ c. $\begin{bmatrix}-\frac{24}{133}\\-\frac{23}{133}\end{bmatrix}$

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Problem 97

Find the inverse of the 2x2 matrix $\left[\begin{array}{cc}-2 & -1 \\ 10 & 7\end{array}\right]$ .

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Problem 98

Find the formula of a linear transformation $\varphi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ given by the matrix $M(\varphi)_{\mathcal{A}}^{\mathcal{B}}=\begin{bmatrix} 1 & 2 \\ 2 & 3 \\ 3 & 1 \end{bmatrix}$ , where $\mathcal{A}=\{(1,0),(1,-1)\}$ and $\mathcal{B}=\{(1,0,0),(0,1,1),(-1,0,1)\}$ . Also, find the matrix $M(\varphi \circ \psi)_{\mathcal{A}}^{\mathcal{B}}$ for the linear transformation $\psi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by $\psi((x_1, x_2))=(x_2, x_1+x_2)$ .

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Problem 99

Find the slope of the line with equation $11x - 4y = 32$ .

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Problem 100

Calculate the difference between vector $\boldsymbol{a}$ and 5 times vector $\boldsymbol{b}$ , and express the result as a column vector.

See Solution

1 2

Linear Algebra

Problem 1

Solve the matrix equation (6143)(xy)=(−35)\left(\begin{array}{ll}6 & 1 \\ 4 & 3\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}-3 \\ 5\end{array}\right)(64​13​)(xy​)=(−35​) for xxx and yyy.

Problem 2

Determine if the given 2×42 \times 42×4 matrix is in row reduced form.

Problem 3

Write an inequality to represent the minimum cargo weight requirement, where xxx is the number of shorter containers (45,600 lbs each) and yyy is the number of longer containers (56,000 lbs each), with a total cargo weight of at least 785,000 lbs.

Problem 4

Solve the system using the given inverse coefficient matrix: x+2y+3z=−3x + 2y + 3z = -3x+2y+3z=−3, x+y+z=4x + y + z = 4x+y+z=4, −x+y+2z=−2-x + y + 2z = -2−x+y+2z=−2.

Problem 5

Write a system of linear inequalities with two lines: a solid line starting at y=1y=1y=1 with slope −5/1-5/1−5/1, and a dashed line starting at y=−2y=-2y=−2 with slope 111. Represent the shaded region between the lines.

Problem 6

Determine if the solution region for the given system of linear inequalities is bounded or unbounded.−x+2y≥4,2x−y≤2,x≥0,y≥0-x + 2y \geq 4, 2x - y \leq 2, x \geq 0, y \geq 0−x+2y≥4,2x−y≤2,x≥0,y≥0

Problem 7

Find the image coordinates of a point Q(−5,1)Q(-5,1)Q(−5,1) under the transformation (x,y)→(−y,−x)(x, y) \rightarrow(-y,-x)(x,y)→(−y,−x).

Problem 8

Problem 9

Find the value of kkk that makes the vectors (2,−6,4)(2,-6,4)(2,−6,4) and (1,k,−2)(1, k,-2)(1,k,−2) collinear.

Problem 10

Determine if the set of all symmetric 2×22 \times 22×2 matrices, WWW, is a subspace of R2×2R^{2 \times 2}R2×2.

Problem 11

Find the eigenvalues of the 2×22 \times 22×2 matrix A=[30−18]A = \left[\begin{array}{cc}3 & 0 \\ -1 & 8\end{array}\right]A=[3−1​08​].

Problem 12

If a set of 3 vectors {v1,v2,v3}\{v_1, v_2, v_3\}{v1​,v2​,v3​} spans R3\mathbb{R}^3R3, then it forms a basis for R3\mathbb{R}^3R3. True or False?

Problem 13

Find the plane equation ax+by+cz=0ax + by + cz = 0ax+by+cz=0 that passes through the points (1,0,2),(−1,1,−2)(1,0,2), (-1,1,-2)(1,0,2),(−1,1,−2), and the origin.

Problem 14

Find the value of ccc if the vectors v1=[1,0,0]v_1 = [1, 0, 0]v1​=[1,0,0], v2=[1,1,0]v_2 = [1, 1, 0]v2​=[1,1,0], and v3=[1,−1,c]v_3 = [1, -1, c]v3​=[1,−1,c] are linearly dependent.

Problem 15

Given constraints and optimal solution (x,y)=(25,0)(x, y) = (25, 0)(x,y)=(25,0), find the values of the slack variables s1s_1s1​ and s2s_2s2​.

Problem 16

Convert 1973.1 cubic miles to cubic kilometers using the conversion ratio: 1.61 km=11.61 \mathrm{~km} = 11.61 km=1 mile.

Problem 17

Problem 18

Finden Sie einen Vektor n⃗\vec{n}n, der senkrecht auf dem Vektor a⃗=(25−3)\vec{a}=\left(\begin{array}{c}2 \\ 5 \\ -3\end{array}\right)a=⎝⎛​25−3​⎠⎞​ steht.

Problem 19

Find the vector −6(−30i+40j)-6(-30\mathbf{i}+40\mathbf{j})−6(−30i+40j).

Problem 20

Find the y-value of the solution to the 2×22 \times 22×2 matrix equation [3221][xy]=[34]\left[\begin{array}{ll} 3 & 2 \\ 2 & 1 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{l} 3 \\ 4 \end{array}\right][32​21​][xy​]=[34​].

Problem 21

Solve for x\mathbf{x}x in terms of a\mathbf{a}a and b\mathbf{b}b using x+6a−b=6(x+a)−(2a−b)\mathbf{x}+6\mathbf{a}-\mathbf{b}=6(\mathbf{x}+\mathbf{a})-(2\mathbf{a}-\mathbf{b})x+6a−b=6(x+a)−(2a−b).

Problem 22

Is the set W={(x1,0,x3):x1,x3 are real numbers}W=\{(x_1, 0, x_3): x_1, x_3 \text{ are real numbers}\}W={(x1​,0,x3​):x1​,x3​ are real numbers} a subspace of R3\mathbb{R}^3R3?

Problem 23

If a square matrix is symmetric, its transpose is also symmetric. True or False?

Problem 24

Find the rank of 2×22 \times 22×2 matrix AAA if Ax=0Ax=0Ax=0 has only the zero solution.

Problem 25

Problem 26

Find the value of bbb where the linear system with augmented matrix [100∣101b∣2024∣4]\left[\begin{array}{lllll}1 & 0 & 0 & \mid & 1 \\ 0 & 1 & b & \mid & 2 \\ 0 & 2 & 4 & \mid & 4\end{array}\right]⎣⎡​100​012​0b4​∣∣∣​124​⎦⎤​ has infinitely many solutions.

Problem 27

Find the values of sss, ttt, and rrr in the given 2×22\times 22×2 matrix equation.

Problem 28

Determine if the lines y=−56x+94y=\frac{-5}{6}x+\frac{9}{4}y=6−5​x+49​ and y=65x+52y=\frac{6}{5}x+\frac{5}{2}y=56​x+25​ are parallel, perpendicular, or neither.

Problem 29

Is it possible for the consistent system Ax=bAx=bAx=b to have multiple solutions for some bbb? Yes, sometimes. No, never.

Problem 30

Convert augmented matrix to system of 3 linear equations in xxx, yyy, and zzz with constant terms.

Problem 31

Find values of rrr and sss that satisfy 3[2r,5s]−[s,3r]=[−18,9]3[2r, 5s] - [s, 3r] = [-18, 9]3[2r,5s]−[s,3r]=[−18,9]. If no solution exists, state "Not Possible".

Problem 32

Find C−A+BC-A+BC−A+B and simplify, if possible. Give exact answers in fraction form, if necessary. Select "Undefined" if applicable.

Problem 33

Choose a system of linear equations to represent the growth of a 444 inch per year spruce tree and a 666 inch per year hemlock tree with initial heights of 141414 inches and 888 inches, respectively.

Problem 34

Solve the system of linear equations represented by the given augmented matrix using Gauss-Jordan method. Perform the row operations and find the values of xxx, yyy, and zzz.

Problem 35

Find the transition matrix from basis FFF to basis EEE where E=[(1,2)T,(3,7)T]E=[(1,2)^{T},(3,7)^{T}]E=[(1,2)T,(3,7)T] and F=[(1,−2)T,(3,−5)T]F=[(1,-2)^{T},(3,-5)^{T}]F=[(1,−2)T,(3,−5)T] in R2\mathbb{R}^{2}R2.

Problem 36

Problem 37

Find zzz using Cramer's rule and a calculator for the system: 4x−y+5z=1,−x+2y+z=0,5x−2y+2z=−44x-y+5z=1, -x+2y+z=0, 5x-2y+2z=-44x−y+5z=1,−x+2y+z=0,5x−2y+2z=−4.

Problem 38

Solve the system Ax=bA \mathbf{x}=\mathbf{b}Ax=b using the given factorization A=P⊤LUA=P^{\top} L UA=P⊤LU, where b=[117]\mathbf{b}=\begin{bmatrix}1 \\ 1 \\ 7\end{bmatrix}b=⎣⎡​117​⎦⎤​ and x=[232]\mathbf{x}=\begin{bmatrix}2 \\ 3 \\ 2\end{bmatrix}x=⎣⎡​232​⎦⎤​.

Problem 39

Find the matrix XXX that satisfies the equation A(X+5B)=CA(X + 5B) = CA(X+5B)=C.

Problem 40

Find the inverse A−1A^{-1}A−1 of the given 3x3 matrix A=[113−2−1−6031]A = \begin{bmatrix} 1 & 1 & 3 \\ -2 & -1 & -6 \\ 0 & 3 & 1 \end{bmatrix}A=⎣⎡​1−20​1−13​3−61​⎦⎤​.

Problem 41

Problem 42

Find the most accurate statement about the given 6×66 \times 66×6 matrix AAA, which could be the adjacency matrix of a graph or a digraph.

Solve the matrix equation $\left(\begin{array}{ll}6 & 1 \\ 4 & 3\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}-3 \\ 5\end{array}\right)$ for $x$ and $y$ .

Determine if the given $2 \times 4$ matrix is in row reduced form.

Write an inequality to represent the minimum cargo weight requirement, where $x$ is the number of shorter containers (45,600 lbs each) and $y$ is the number of longer containers (56,000 lbs each), with a total cargo weight of at least 785,000 lbs.

Solve the system using the given inverse coefficient matrix: $x + 2y + 3z = -3$ , $x + y + z = 4$ , $-x + y + 2z = -2$ .

Write a system of linear inequalities with two lines: a solid line starting at $y=1$ with slope $-5/1$ , and a dashed line starting at $y=-2$ with slope $1$ . Represent the shaded region between the lines.

Determine if the solution region for the given system of linear inequalities is bounded or unbounded.
$-x + 2y \geq 4, 2x - y \leq 2, x \geq 0, y \geq 0$

Find the image coordinates of a point $Q(-5,1)$ under the transformation $(x, y) \rightarrow(-y,-x)$ .

Find the value of $k$ that makes the vectors $(2,-6,4)$ and $(1, k,-2)$ collinear.

Determine if the set of all symmetric $2 \times 2$ matrices, $W$ , is a subspace of $R^{2 \times 2}$ .

Find the eigenvalues of the $2 \times 2$ matrix $A = \left[\begin{array}{cc}3 & 0 \\ -1 & 8\end{array}\right]$ .

If a set of 3 vectors $\{v_1, v_2, v_3\}$ spans $\mathbb{R}^3$ , then it forms a basis for $\mathbb{R}^3$ . True or False?

Find the plane equation $ax + by + cz = 0$ that passes through the points $(1,0,2), (-1,1,-2)$ , and the origin.

Find the value of $c$ if the vectors $v_1 = [1, 0, 0]$ , $v_2 = [1, 1, 0]$ , and $v_3 = [1, -1, c]$ are linearly dependent.

Given constraints and optimal solution $(x, y) = (25, 0)$ , find the values of the slack variables $s_1$ and $s_2$ .

Convert 1973.1 cubic miles to cubic kilometers using the conversion ratio: $1.61 \mathrm{~km} = 1$ mile.

Finden Sie einen Vektor $\vec{n}$ , der senkrecht auf dem Vektor $\vec{a}=\left(\begin{array}{c}2 \\ 5 \\ -3\end{array}\right)$ steht.

Find the vector $-6(-30\mathbf{i}+40\mathbf{j})$ .

Find the y-value of the solution to the $2 \times 2$ matrix equation $\left[\begin{array}{ll} 3 & 2 \\ 2 & 1 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{l} 3 \\ 4 \end{array}\right]$ .

Solve for $\mathbf{x}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ using $\mathbf{x}+6\mathbf{a}-\mathbf{b}=6(\mathbf{x}+\mathbf{a})-(2\mathbf{a}-\mathbf{b})$ .

Is the set $W=\{(x_1, 0, x_3): x_1, x_3 \text{ are real numbers}\}$ a subspace of $\mathbb{R}^3$ ?

Find the rank of $2 \times 2$ matrix $A$ if $Ax=0$ has only the zero solution.

Find the value of $b$ where the linear system with augmented matrix $\left[\begin{array}{lllll}1 & 0 & 0 & \mid & 1 \\ 0 & 1 & b & \mid & 2 \\ 0 & 2 & 4 & \mid & 4\end{array}\right]$ has infinitely many solutions.

Find the values of $s$ , $t$ , and $r$ in the given $2\times 2$ matrix equation.

Determine if the lines $y=\frac{-5}{6}x+\frac{9}{4}$ and $y=\frac{6}{5}x+\frac{5}{2}$ are parallel, perpendicular, or neither.

Is it possible for the consistent system $Ax=b$ to have multiple solutions for some $b$ ? Yes, sometimes. No, never.

Convert augmented matrix to system of 3 linear equations in $x$ , $y$ , and $z$ with constant terms.

Find values of $r$ and $s$ that satisfy $3[2r, 5s] - [s, 3r] = [-18, 9]$ . If no solution exists, state "Not Possible".

Find $C-A+B$ and simplify, if possible. Give exact answers in fraction form, if necessary. Select "Undefined" if applicable.

Choose a system of linear equations to represent the growth of a $4$ inch per year spruce tree and a $6$ inch per year hemlock tree with initial heights of $14$ inches and $8$ inches, respectively.

Solve the system of linear equations represented by the given augmented matrix using Gauss-Jordan method. Perform the row operations and find the values of $x$ , $y$ , and $z$ .

Find the transition matrix from basis $F$ to basis $E$ where $E=[(1,2)^{T},(3,7)^{T}]$ and $F=[(1,-2)^{T},(3,-5)^{T}]$ in $\mathbb{R}^{2}$ .

Find $z$ using Cramer's rule and a calculator for the system: $4x-y+5z=1, -x+2y+z=0, 5x-2y+2z=-4$ .

Solve the system $A \mathbf{x}=\mathbf{b}$ using the given factorization $A=P^{\top} L U$ , where $\mathbf{b}=\begin{bmatrix}1 \\ 1 \\ 7\end{bmatrix}$ and $\mathbf{x}=\begin{bmatrix}2 \\ 3 \\ 2\end{bmatrix}$ .

Find the matrix $X$ that satisfies the equation $A(X + 5B) = C$ .

Find the inverse $A^{-1}$ of the given 3x3 matrix $A = \begin{bmatrix} 1 & 1 & 3 \\ -2 & -1 & -6 \\ 0 & 3 & 1 \end{bmatrix}$ .

Find the most accurate statement about the given $6 \times 6$ matrix $A$ , which could be the adjacency matrix of a graph or a digraph.

Find the intersection point of 3 one-way streets using Gaussian elimination to solve the system of $x+9=y+13$ , $z+15=x+10$ , $y+23=z+24$ .

Find a vector $\mathbf{v}$ with magnitude 4, where the $\mathbf{i}$ component is twice the $\mathbf{j}$ component.

Given the $3 \times 4$ matrix $D$ , find the value of the element $d_{23}$ . If the element is undefined, press the undefined button.

Find the least squares line for the set of points $(x, y)$ where $x \in [1, 10]$ and $y = [0.4, 0.9, 1.4, 2.4, 2.9, 3.4, 3.4, 4.4, 4.9, 5.4]$ .

Finde die Funktionsgleichung einer linearen Funktion mit Steigung $m=-2$ , die durch den Punkt $\mathbf{P (-7/-5)}$ verläuft.

Solve the system of 3 linear equations in 3 variables: $-3x+4y+2z=-28, x+5y-4z=-6, 4x+y-6z=22$ .

Perform matrix operations on $A$ , $B$ , $C$ , and $D$ matrices, such as $B+D$ and $B-A$ .

Compute the product of the given $3 \times 1$ and $1 \times 3$ matrices.

Find the value of the $3 \times 3$ determinant: $\left|\begin{array}{ccc} 3 & -3 & 0 \\ 1 & -2 & 1 \\ -2 & 1 & 2 \end{array}\right|$

Solve the system of linear equations using Gaussian elimination: $2x + y - z = 2, x + 3y + 2z = 1, x + y + z = 2$ .

Find the determinant of the matrix product of $A=\left[\begin{array}{ll}4 & 2\end{array}\right]$ and $B=\left[\begin{array}{c}-1 \\ 3\end{array}\right]$ .

Find the value(s) of $a$ such that the $3\\times 3$ matrix $A=\left[\begin{array}{lll}2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 1 & a\end{array}\right]$ has no inverse.

Find the matrix product of -3 and a 2x2 matrix $E=\begin{bmatrix}-4 & 5 \\ 6 & -1\end{bmatrix}$ .

Find the product of matrix $C$ with $x$ and matrix $D$ with $y$ , then add the results.

Find the min and max of $z=3x+9y$ subject to $3x+4y \geq 12$ , $x+4y \geq 8$ , $x \geq 0$ , $y \geq 0$ . (Round min to nearest tenth.)

Find the product of two $2 \times 2$ matrices $X$ and $Y$ , where $X=\left[\begin{array}{cc}4 & 4 \\ -3 & -1\end{array}\right]$ and $Y=\left[\begin{array}{cc}4 & 1 \\ 4 & -1\end{array}\right]$ .

Find vectors parallel to $\left(\begin{array}{c}-2 \\ 4\end{array}\right)$ .

Maximize linear function $p = x - 2y$ subject to linear constraints. Determine if function is unbounded.

Use Cramer's rule to find the value of $y$ that satisfies the given system of 3 linear equations in 3 variables.

Solve the linear programming problem using the simplex method to maximize $P=x+4y-2z$ subject to $3x+y-z\leq44$ , $2x+y-z\leq22$ , $-x+y+z\leq44$ , and $x,y,z\geq0$ .

Write a system of linear equations from the given augmented matrix, then solve using back-substitution. Variables: $x, y, z$ .

Perform the given row operations on the provided $2 \times 3$ matrix.

Find $\frac{1}{3} B+2 A$ given $A=\left[\begin{array}{ll}6 & 3\end{array}\right]$ and $B=\left[\begin{array}{ll}-15 & 6\end{array}\right]$ .

Find the difference between two 2x2 matrices $A$ and $B$ .

Find the product of two $2 \times 2$ matrices: $\left[\begin{array}{cc}-2 & 0\\0 & -4\\0 & 3\\-2 & 0\end{array}\right]$ and $\left[\begin{array}{cc}-1 & -1\\3 & -1\end{array}\right]$ .