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) for xx and yy.

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Problem 2

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

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Problem 3

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

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Problem 4

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

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Problem 5

Write a system of linear inequalities with two lines: a solid line starting at y=1y=1 with slope 5/1-5/1, and a dashed line starting at y=2y=-2 with slope 11. 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+2y4,2xy2,x0,y0-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)Q(-5,1) under the transformation (x,y)(y,x)(x, y) \rightarrow(-y,-x).

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Problem 8

Fill in the missing values in the matrix equation [100010001][000]=[???]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} ? \\ ? \\ ? \end{bmatrix}.

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Problem 9

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

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Problem 10

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

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Problem 11

Find the eigenvalues of the 2×22 \times 2 matrix A=[3018]A = \left[\begin{array}{cc}3 & 0 \\ -1 & 8\end{array}\right].

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Problem 12

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

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Problem 13

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

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Problem 14

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

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Problem 15

Given constraints and optimal solution (x,y)=(25,0)(x, y) = (25, 0), find the values of the slack variables s1s_1 and s2s_2.

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Problem 16

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

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Problem 17

Solve for matrix XX in (A5B)X=C(A-5B)X=C given A=[5012],B=[1203],C=[2635]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].

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Problem 18

Finden Sie einen Vektor n\vec{n}, der senkrecht auf dem Vektor a=(253)\vec{a}=\left(\begin{array}{c}2 \\ 5 \\ -3\end{array}\right) steht.

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Problem 19

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

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Problem 20

Find the y-value of the solution to the 2×22 \times 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].

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Problem 21

Solve for x\mathbf{x} in terms of a\mathbf{a} and b\mathbf{b} using x+6ab=6(x+a)(2ab)\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={(x1,0,x3):x1,x3 are real numbers}W=\{(x_1, 0, x_3): x_1, x_3 \text{ are real numbers}\} a subspace of R3\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 2×22 \times 2 matrix AA if Ax=0Ax=0 has only the zero solution.

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Problem 25

Solve linear system Ax=bAx=b where AA is 3×33 \times 3 matrix, x=[x1x2x3]x=\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right], b=[243]b=\left[\begin{array}{l}-2 \\ -4 \\ -3\end{array}\right], and A1=[141232054]A^{-1}=\left[\begin{array}{ccc}1 & 4 & -1 \\ 2 & -3 & 2 \\ 0 & 5 & -4\end{array}\right]. Find the value of x3x_{3}.

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Problem 26

Find the value of bb where the linear system with augmented matrix [100101b20244]\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 ss, tt, and rr in the given 2×22\times 2 matrix equation.

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Problem 28

Determine if the lines y=56x+94y=\frac{-5}{6}x+\frac{9}{4} and y=65x+52y=\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=bAx=b to have multiple solutions for some bb? Yes, sometimes. No, never.

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Problem 30

Convert augmented matrix to system of 3 linear equations in xx, yy, and zz with constant terms.

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Problem 31

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

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Problem 32

Find CA+BC-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 44 inch per year spruce tree and a 66 inch per year hemlock tree with initial heights of 1414 inches and 88 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 xx, yy, and zz.

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Problem 35

Find the transition matrix from basis FF to basis EE where 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}] in R2\mathbb{R}^{2}.

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Problem 36

Solve the linear system Ax=bAx=b where AA is a 3×33\times 3 matrix, x=[x1x2x3]x=\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}, and b=[505]b=\begin{bmatrix}5\\0\\-5\end{bmatrix}. If A1=[141232054]A^{-1}=\begin{bmatrix}1&4&-1\\2&-3&2\\0&5&-4\end{bmatrix}, then x3=x_3=

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Problem 37

Find zz using Cramer's rule and a calculator for the system: 4xy+5z=1,x+2y+z=0,5x2y+2z=44x-y+5z=1, -x+2y+z=0, 5x-2y+2z=-4.

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Problem 38

Solve the system Ax=bA \mathbf{x}=\mathbf{b} using the given factorization A=PLUA=P^{\top} L U, where b=[117]\mathbf{b}=\begin{bmatrix}1 \\ 1 \\ 7\end{bmatrix} and x=[232]\mathbf{x}=\begin{bmatrix}2 \\ 3 \\ 2\end{bmatrix}.

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Problem 39

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

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Problem 40

Find the inverse A1A^{-1} of the given 3x3 matrix A=[113216031]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) 4x13x2+6x3=0,2x1x3=5,4x1=24x_1 - 3x_2 + 6x_3 = 0, 2x_1 - x_3 = 5, 4x_1 = -2, (b) 4x1x2+3x3=2,x1+3x2=5,4x2=84x_1 - x_2 + 3x_3 = 2, x_1 + 3x_2 = 5, 4x_2 = 8, (c) 5x1=10,5x23x3=9,4x1+x2=05x_1 = 10, 5x_2 - 3x_3 = 9, 4x_1 + x_2 = 0, (d) a+b=3,a+bc=0,b+c=4a + b = 3, a + b - c = 0, b + c = 4, (e) r+st=0,r+t=2,r2s+t=2r + s - t = 0, r + t = 2, r - 2s + t = 2, (f) 0.6y+1.8z=3,0.3x+1.2y=0,0.5x+z=10.6y + 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 6×66 \times 6 matrix AA, 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=3x1+7x2+8x3z=3x_1+7x_2+8x_3 subject to 5x1x2+x31500,2x1+2x2+x32500,4x1+2x2+x32000,x1,x2,x305x_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+13x+9=y+13, z+15=x+10z+15=x+10, y+23=z+24y+23=z+24.

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Problem 46

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

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Problem 47

Determine if 3×33 \times 3 matrices A=[312010]A = \left[\begin{array}{rr} 3 & -1 \\ -20 & 10 \end{array}\right] and B=[11102310]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 AA in the equation A[066122078]=[910228656]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=[000000000]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 3×43 \times 4 matrix DD, find the value of the element d23d_{23}. If the element is undefined, press the undefined button.

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Problem 50

Find the value(s) of kk that satisfy the matrix equation [22k][130305051][22k]=0\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)(x, y) where x[1,10]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]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=2m=-2, die durch den Punkt P(7/5)\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+5y4z=6,4x+y6z=22-3x+4y+2z=-28, x+5y-4z=-6, 4x+y-6z=22.

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Problem 54

Perform matrix operations on AA, BB, CC, and DD matrices, such as B+DB+D and BAB-A.

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Problem 55

Compute the product of the given 3×13 \times 1 and 1×31 \times 3 matrices.

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Problem 56

Find the value of the 3×33 \times 3 determinant: 330121212\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+yz=2,x+3y+2z=1,x+y+z=22x + y - z = 2, x + 3y + 2z = 1, x + y + z = 2.

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Problem 58

Which linear transformation from R3\mathbb{R}^{3} to R2\mathbb{R}^{2} is defined by T([x1x2x3])=[x1x2]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=[42]A=\left[\begin{array}{ll}4 & 2\end{array}\right] and B=[13]B=\left[\begin{array}{c}-1 \\ 3\end{array}\right].

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Problem 60

Find the value(s) of aa such that the 3times33\\times 3 matrix A=[20111011a]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=[4561]E=\begin{bmatrix}-4 & 5 \\ 6 & -1\end{bmatrix}.

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Problem 62

Find the basis and dimension of the subspace VR4V \subset \mathbb{R}^{4} given by the homogeneous system x1+2x2+x3+2x4=0,3x1+5x2+4x3+7x4=0x_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 WR4W \subset \mathbb{R}^{4} spanned by the basis of VV and the vector w=(1,0,0,0)w = (1, 0, 0, 0).

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Problem 63

Find the product of matrix CC with xx and matrix DD with yy, then add the results.

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Problem 64

Find the min and max of z=3x+9yz=3x+9y subject to 3x+4y123x+4y \geq 12, x+4y8x+4y \geq 8, x0x \geq 0, y0y \geq 0. (Round min to nearest tenth.)

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Problem 65

Find the product of two 2×22 \times 2 matrices XX and YY, where X=[4431]X=\left[\begin{array}{cc}4 & 4 \\ -3 & -1\end{array}\right] and Y=[4141]Y=\left[\begin{array}{cc}4 & 1 \\ 4 & -1\end{array}\right].

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Problem 66

Find vectors parallel to (24)\left(\begin{array}{c}-2 \\ 4\end{array}\right).

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Problem 67

Maximize linear function p=x2yp = 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 yy 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+4y2zP=x+4y-2z subject to 3x+yz443x+y-z\leq44, 2x+yz222x+y-z\leq22, x+y+z44-x+y+z\leq44, and x,y,z0x,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,zx, y, z.

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Problem 71

Perform the given row operations on the provided 2×32 \times 3 matrix.

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Problem 72

Find 13B+2A\frac{1}{3} B+2 A given A=[63]A=\left[\begin{array}{ll}6 & 3\end{array}\right] and B=[156]B=\left[\begin{array}{ll}-15 & 6\end{array}\right].

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Problem 73

Find the difference between two 2x2 matrices AA and BB.

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Problem 74

Find the product of two 2×22 \times 2 matrices: [20040320]\left[\begin{array}{cc}-2 & 0\\0 & -4\\0 & 3\\-2 & 0\end{array}\right] and [1131]\left[\begin{array}{cc}-1 & -1\\3 & -1\end{array}\right].

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Problem 75

Rewrite rotation formulas to eliminate xyx'y'-term in equation x2+16xy+y23=0x^2 + 16xy + y^2 - 3 = 0.

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Problem 76

Find the value of the objective function z=5x+8yz=5x+8y at the corner points of the region where x0,y0,2x+y12,x+y6x\geq 0, y\geq 0, 2x+y\leq 12, x+y\geq 6.

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Problem 77

Determine the value of d23d_{23} in the given 2×32 \times 3 matrix DD. If d23d_{23} is not present, state "None".

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Problem 78

Find the vector 4v4\mathbf{v} given v=6,0\mathbf{v}=\langle-6,0\rangle.

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Problem 79

Find the dimension and value of the matrix element a12a_{12}.

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Problem 80

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

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Problem 81

Find the closest vector x\mathbf{x} in the subspace WW spanned by v1\mathbf{v}_{1} and v2\mathbf{v}_{2} to the given vector y\mathbf{y}, where y=[11119]\mathbf{y}=\left[\begin{array}{r}1 \\ -1 \\ 1 \\ 19\end{array}\right], v1=[1111]\mathbf{v}_{1}=\left[\begin{array}{r}1 \\ -1 \\ -1 \\ 1\end{array}\right], and v2=[1102]\mathbf{v}_{2}=\left[\begin{array}{r}-1 \\ 1 \\ 0 \\ 2\end{array}\right].

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Problem 82

Find a symmetric matrix AA with non-negative eigenvalues and show that AA has a symmetric square root RR such that R2=AR^2 = A. Then, find two different square roots of the matrix B=[5335]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 u=2i3j\mathbf{u} = 2\mathbf{i} - 3\mathbf{j} and v=5j\mathbf{v} = 5\mathbf{j}.

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Problem 84

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

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Problem 85

Minimize c=x+2yc=x+2y subject to x+5y19,4x+y19,x0,y0x+5y \geq 19, 4x+y \geq 19, x \geq 0, y \geq 0, where c=(x,y)=(x)c = (x,y)=(\sqrt{x}).

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Problem 86

Compute the product of two 3x3 matrices: [004222025]\left[\begin{array}{ccc}0 & 0 & 4 \\ -2 & -2 & 2 \\ 0 & -2 & -5\end{array}\right] and [111130011]\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 DD, DxD_x, and DyD_y.
6(x+y)=9x+66x=12y12 6(x+y) = 9x + 6 \\ 6x = 12y - 12
D=Dx=Dy= 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=1x + y + 2z = 1, xy=1x - y = 1, and xz=2x - z = 2.

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Problem 89

Find the maximum value of z=4x+5yz=4x+5y subject to the constraints 3x+4y403x+4y\leq40, 7x+3y497x+3y\leq49, and y0y\geq0 using the Desmos Graphing Calculator 2.

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Problem 90

Find the determinant of the 3×33 \times 3 matrix with entries 1,1,0;1,0,2;1,3,1-1, 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 3-3 when x1=115x_1=\frac{11}{5}, x2=0x_2=0, x3=114x_3=\frac{11}{4}, s1=112s_1=\frac{11}{2}, and s2=0s_2=0.

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Problem 92

Zadanie 5. Given A={(1,4),(1,5)},B={(2,1),(5,2)}\mathcal{A}=\{(1,4),(1,5)\}, \mathcal{B}=\{(2,1),(5,2)\} and linear transformation φ:R2R2\varphi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} with M(φ)AB=[1023]M(\varphi)_{\mathcal{A}}^{\mathcal{B}}=\left[\begin{array}{ll}1 & 0 \\ 2 & 3\end{array}\right], find M(φφ)ABM(\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)}A = \{(2,1), (2,3), (-1,4), (1,1)\}.

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Problem 94

Find the values of aa for which the 2×22 \times 2 matrix [2a433a]\begin{bmatrix} 2-a & 4 \\ 3 & 3-a \end{bmatrix} is singular.

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Problem 95

Find the absolute value of 3A3A if AA is a 3×33 \times 3 matrix and 89= ⁣2AT ⁣\frac{8}{9} = |\!2A^T\!|.

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Problem 96

Find the least squares solution to Ax=bAx=b where A=[351701]A=\begin{bmatrix}3&-5\\1&-7\\0&1\end{bmatrix} and b=[025]b=\begin{bmatrix}0\\2\\5\end{bmatrix}. Options: a. [2426623266]\begin{bmatrix}-\frac{24}{266}\\\frac{23}{266}\end{bmatrix} b. [2423]\begin{bmatrix}-24\\-23\end{bmatrix} c. [2413323133]\begin{bmatrix}-\frac{24}{133}\\-\frac{23}{133}\end{bmatrix}

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Problem 97

Find the inverse of the 2x2 matrix [21107]\left[\begin{array}{cc}-2 & -1 \\ 10 & 7\end{array}\right].

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Problem 98

Find the formula of a linear transformation φ:R2R3\varphi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3} given by the matrix M(φ)AB=[122331]M(\varphi)_{\mathcal{A}}^{\mathcal{B}}=\begin{bmatrix} 1 & 2 \\ 2 & 3 \\ 3 & 1 \end{bmatrix}, where A={(1,0),(1,1)}\mathcal{A}=\{(1,0),(1,-1)\} and B={(1,0,0),(0,1,1),(1,0,1)}\mathcal{B}=\{(1,0,0),(0,1,1),(-1,0,1)\}. Also, find the matrix M(φψ)ABM(\varphi \circ \psi)_{\mathcal{A}}^{\mathcal{B}} for the linear transformation ψ:R2R2\psi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} given by ψ((x1,x2))=(x2,x1+x2)\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 11x4y=3211x - 4y = 32.

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Problem 100

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

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