?In Exercises 27-28, how many \(3 \times 3\) matrices A can you find for which the

In Exercises 1-2, suppose that \(A, B, C, D, \text { and } E\) are matrices with the following sizes: \(A(4 \times 5)\) \(B(4 \times 5)\) \(C(5 \times 2)\) \(D(4 \times 2)\) \(E(5 \times 4)\) In each part, determine whether the given matrix expression is defined. For those that are defined, give the size of the resulting matrix a. \(B A\) b. \(A B^{T}\) c. \(A C+D\) d. \(E(A C)\) e. \(A-3 E^{T}\) f. \(E(5 B+A)\) Equation Transcription: Text Transcription: A,B,C,D, and E A (4 x 5) B (4 x 5) C (5 x 2) D (4 x 2) E (5 x 4) BA AB^T AC+D E(AC) A-3E^T E(5B+A)

In Exercises 1-2, suppose that \(A, B, C, D, \text { and } E\) are matrices with the following sizes: \(A(4 \times 5)\) \(B(4 \times 5)\) \(C(5 \times 2)\) \(D(4 \times 2)\) \(E(5 \times 4)\) In each part, determine whether the given matrix expression is defined. For those that are defined, give the size of the resulting matrix 1. \(C D^{T}\) 2. \(D C\) 3. \(B C-3 D\) 4. \(D^{T}(B E)\) 5. \(B^{T} D+E D\) 6. \(B A^{T}+D\) Equation Transcription: Text Transcription: A,B,C,D, and E A (4 x 5) B (4 x 5) C (5 x 2) D (4 x 2) E (5 x 4) CD^T DC BC-3D D^T(BE) B^T+ED BA^T+D

In Exercises 3-6, use the following matrices to compute the indicated expression if it is defined. \(A=\left[\begin{array}{rr} 3 & 0 \\ -1 & 2 \\ 1 & 1 \end{array}\right]\) \(B=\left[\begin{array}{ll} 4 & -1 \\ 0 & 2 \end{array}\right]\) \(C=\left[\begin{array}{lll} 1 & 4 & 2 \\ 3 & 1 & 5 \end{array}\right]\) \(D=\left[\begin{array}{rrr} 1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4 \end{array}\right]\) \(E=\left[\begin{array}{lll} 6 & 1 & 3 \\ -1 & 1 & 2 \\ 4 & 1 & 3 \end{array}\right]\) a. \(D+E\) b. \(D-E\) c. \(5 A\) d. \(-7 C\) e. \(2 B-C\) f. \(4 E-2 D\) g. \(-3(D+2 E)\) h. \(A-A\) i. \(\operatorname{tr}(D)\) j. \(\operatorname{tr}(D-3 E)\) k. \(4 \operatorname{tr}(7 B)\) l. \(\operatorname{tr}(A)\) Equation Transcription: [] [] [] [] [] Text Transcription: A=[ 3 & 0 \\ -1 & 2 \\ 1 & 1 ] B=[ 4 & -1 \\ 0 & 2 ] C=[ 1 & 4 & 2 \\ 3 & 1 & 5 ] D=[ 1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4 ] E=[ 6 & 1 & 3 \\ -1 & 1 & 2 \\ 4 & 1 & 3 ] D+E D-E 5A -7C 2B-C 4E-2D -3(D+2E) A-A tr(D) tr(D-3E) 4tr(7B) tr(A)

In Exercises 3-6, use the following matrices to compute the indicated expression if it is defined. \(A=\left[\begin{array}{rr} 3 & 0 \\ -1 & 2 \\ 1 & 1 \end{array}\right]\) \(B=\left[\begin{array}{ll} 4 & -1 \\ 0 & 2 \end{array}\right]\) \(C=\left[\begin{array}{lll} 1 & 4 & 2 \\ 3 & 1 & 5 \end{array}\right]\) \(D=\left[\begin{array}{rrr} 1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4 \end{array}\right]\) \(E=\left[\begin{array}{lll} 6 & 1 & 3 \\ -1 & 1 & 2 \\ 4 & 1 & 3 \end{array}\right]\) a. \(2 A^{T}+C\) b. \(D^{T}-E^{T}\) c. \((D-E)^{T}\) d. \(B^{T}+5 C^{T}\) e. \(\frac{1}{2} C^{T}-\frac{1}{4} A\) f. \(B-B^{T}\) g. \(2 E^{T}-3 D^{T}\) H. \(\left(2 E^{T}-3 D^{T}\right)^{T}\) i. \((C D) E\) j. \(C(B A)\) k. \(\operatorname{tr}\left(D E^{T}\right)\) l. \(\operatorname{tr}(B C)\) Equation Transcription: [] [] [] [] [] Text Transcription: A=[ 3 & 0 \\ -1 & 2 \\ 1 & 1 ] B=[ 4 & -1 \\ 0 & 2 ] C=[ 1 & 4 & 2 \\ 3 & 1 & 5 ] D=[ 1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4 ] E=[ 6 & 1 & 3 \\ -1 & 1 & 2 \\ 4 & 1 & 3 ] 2A^T+C D^T-E^T (D-E)^T B^T+5C^T 12C^T-14A B-B^T 2E^T-3D^T (CD)E C(BA) tr(DE^T) tr(BC)

In Exercises 3-6, use the following matrices to compute the indicated expression if it is defined. \(A=\left[\begin{array}{rr} 3 & 0 \\ -1 & 2 \\ 1 & 1 \end{array}\right]\) \(B=\left[\begin{array}{ll} 4 & -1 \\ 0 & 2 \end{array}\right]\) \(C=\left[\begin{array}{lll} 1 & 4 & 2 \\ 3 & 1 & 5 \end{array}\right]\) \(D=\left[\begin{array}{rrr} 1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4 \end{array}\right]\) \(E=\left[\begin{array}{lll} 6 & 1 & 3 \\ -1 & 1 & 2 \\ 4 & 1 & 3 \end{array}\right]\) a. \(A B\) b. \(B A\) c. \((3 E) D\) d. \((A B) C\) e. \(A(B C)\) f. \(C C^{T}\) g. \((D A)^{T}\) h. \(\left(C^{T} B\right) A^{T}\) i. \(\operatorname{tr}\left(D D^{T}\right)\) j. \(\operatorname{tr}\left(4 E^{T}-D\right)\) k. \(\operatorname{tr}\left(C^{T} A^{T}+2 E^{T}\right)\) l. \(\operatorname{tr}\left(\left(E C^{T}\right)^{T} A\right)\) Equation Transcription: [] [] [] [] [] Text Transcription: A=[ 3 & 0 \\ -1 & 2 \\ 1 & 1 ] B=[ 4 & -1 \\ 0 & 2 ] C=[ 1 & 4 & 2 \\ 3 & 1 & 5 ] D=[ 1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4 ] E=[ 6 & 1 & 3 \\ -1 & 1 & 2 \\ 4 & 1 & 3 ] AB BA (3E)D (AB)C A(BC) CCT (DA)T (CTB)AT tr(DDT) tr(4ET-D) tr(CTAT +2ET) tr( (ECT)TA)

In Exercises 3-6, use the following matrices to compute the indicated expression if it is defined. \(A=\left[\begin{array}{rr} 3 & 0 \\ -1 & 2 \\ 1 & 1 \end{array}\right]\) \(B=\left[\begin{array}{ll} 4 & -1 \\ 0 & 2 \end{array}\right]\) \(C=\left[\begin{array}{lll} 1 & 4 & 2 \\ 3 & 1 & 5 \end{array}\right]\) \(D=\left[\begin{array}{rrr} 1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4 \end{array}\right]\) \(E=\left[\begin{array}{lll} 6 & 1 & 3 \\ -1 & 1 & 2 \\ 4 & 1 & 3 \end{array}\right]\) a. \(\left(2 D^{T}-E\right) A\) b. \((4 B) C+2 B\) c. \((-A C)^{T}+5 D^{T}\) d. \(\left(B A^{T}-2 C\right)^{T}\) e. \(B^{T}\left(C C^{T}-A^{T} A\right)\) f. \(D^{T} E^{T}-(E D)^{T}\) Equation Transcription: [] [] [] [] [] Text Transcription: A=[ 3 & 0 \\ -1 & 2 \\ 1 & 1 ] B=[ 4 & -1 \\ 0 & 2 ] C=[ 1 & 4 & 2 \\ 3 & 1 & 5 ] D=[ 1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4 ] E=[ 6 & 1 & 3 \\ -1 & 1 & 2 \\ 4 & 1 & 3 ] (2D^T-E)A (4B)C+2B (-AC)^T+5D^T (BA^T-2C)^T B^T(CC^T-A^TA) D^TE^T-(ED)^T

In Exercises 7-8, use the following matrices and either the row method or the column method, as appropriate, to find the indicated row or column. \(A=\left[\begin{array}{ccc} 3 & -2 & 7 \\ 6 & 5 & 4 \\ 0 & 4 & 9 \end{array}\right]\) \(B=\left[\begin{array}{lll} 6 & -2 & 4 \\ 0 & 1 & 3 \\ 7 & 7 & 5 \end{array}\right]\) a. the first row of \(A B\) b. the third row of \(A B\) c. the second column of \(A B\) d. the first column of \(B A\) e. the third row of \(A A\) f. the third column of \(A A\) Equation Transcription: [] [] Text Transcription: A=[ 3 & -2 & 7 \\ 6 & 5 & 4 \\ 0 & 4 & 9 ] B=[ 6 & -2 & 4 \\ 0 & 1 & 3 \\ 7 & 7 & 5 ] AB AB AB BA AA AA

In Exercises 7-8, use the following matrices and either the row method or the column method, as appropriate, to find the indicated row or column. \(A=\left[\begin{array}{ccc} 3 & -2 & 7 \\ 6 & 5 & 4 \\ 0 & 4 & 9 \end{array}\right]\) \(B=\left[\begin{array}{lll} 6 & -2 & 4 \\ 0 & 1 & 3 \\ 7 & 7 & 5 \end{array}\right]\) a. the first column of \(\overrightarrow{A B}\) b. the third column of \(B B\) c. the second row of \(B B\) d. the first column of \(A A\) e. the third column of \(A B\) f. the first row of \(B A\) Equation Transcription: [] [] Text Transcription: A=[ 3 & -2 & 7 \\ 6 & 5 & 4 \\ 0 & 4 & 9 ] B=[ 6 & -2 & 4 \\ 0 & 1 & 3 \\ 7 & 7 & 5 ] \overrightarrowA B BB BB AA AB BA

In Exercises 9-10, use matrices \(A \text { and } B\) from Exercises 7-8 a. Express each column vector of \(A A\) as a linear combination of the column vectors of \(A\). b. Express each column vector of \(B B\) as a linear combination of the column vectors of \(B\). Equation Transcription: Text Transcription: A and B AA A BB B

In Exercises 9-10, use matrices \(A \text { and } B\) from Exercises 7-8 a. Express each column vector of \(A B\) as a linear combination of the column vectors of \(A\). b. Express each column vector of \(B A\) as a linear combination of the column vectors of \(B\). Equation Transcription: Text Transcription: A and B AB A BA B

In each part of Exercises 11-12, find matrices \(A, x, \text { and } b\) that express the given linear system as a single matrix equation \(A x=b\), and write out this matrix equation. a. \(2 x_{1}-3 x_{2}+5 x_{3}=7\) \(9 x_{1}-x_{2}+x_{3}=-1\) \(x_{1}+5 x_{2}+4 x_{3}=0\) b. \(4 x_{1}-3 x_{3}+x_{4}=1\) \(5 x_{1}+x_{2}-8 x_{4}=3\) \(2 x_{1}-5 x_{2}+9 x_{3}-x_{4}=0\) \(3 x_{2}-x_{3}+7 x_{4}=2\) Equation Transcription: Text Transcription: A,x and b Ax=b 2x_1-3x_2+5x_3=7 9x_1-x_2+x_3=-1 x_1+5x_2+4x_3=0 4x_1-3x_3+x_4=1 5x_1+x_2-8x_4=3 2x_1-5x_2+9x_3-x_4=0 3x_2-x_3+7x_4=2

In each part of Exercises 11-12, find matrices \(A, x, \text { and } b\) that express the given linear system as a single matrix equation \(A x=b\), and write out this matrix equation. a. \(x_{1}-2 x_{2}+3 x_{3}=-3\) \(2 x_{1}+x_{2}=0\) \(-3 x_{2}+4 x_{3}=1\) \(x_{1}+x_{3}=5\) b. \(3 x_{1}+3 x_{2}+3 x_{3}=-3\) \(-x_{1}-5 x_{2}-2 x_{3}=3\) \(-4 x_{2}+x_{3}=0\) Equation Transcription: Text Transcription: A,x and b Ax=b x_1-2x_2+3x_3=-3 2x_1+x_2=0 -3x_2+4x_3=1 x_1+x_3=5 3x_1+3x_2+3x_3=-3 -x_1-5x_2-2x_3=3 -4x_2+x_3=0

In each part of Exercises 13-14, express the matrix equation as a system of linear equations. a. \(\left[\begin{array}{ccc} 5 & 6 & -7 \\ -1 & -2 & 3 \\ 0 & 4 & -1 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]=\left[\begin{array}{l} 2 \\ 0 \\ 3 \end{array}\right]\) b. \(\left[\begin{array}{rrr} 1 & 1 & 1 \\ 2 & 3 & 0 \\ 5 & -3 & -6 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 2 \\ 2 \\ -9 \end{array}\right]\) Equation Transcription: [] [] [] [] Text Transcription: [ 5 & 6 & -7 \\ -1 & -2 & 3 \\ 0 & 4 & -1] [ x_1 \ x_2 \ x_3 ]=[ 2 \ 0 \ 3 ] [ 1 & 1 & 1 \\ 2 & 3 & 0 \\ 5 & -3 & -6 ][ x \\ y \\ z ]=[ 2 \\ 2 \\ -9 ]

In each part of Exercises 13-14, express the matrix equation as a system of linear equations. a. \(\left[\begin{array}{rcl} 3 & -1 & 2 \\ 4 & 3 & 7 \\ -2 & 1 & 5 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]=\left[\begin{array}{c} 2 \\ -1 \\ 4 \end{array}\right]\) b. \(\left[\begin{array}{rrrr} 3 & -2 & 0 & 1 \\ 5 & 0 & 2 & -2 \\ 3 & 1 & 4 & 7 \\ -2 & 5 & 1 & 6 \end{array}\right]\left[\begin{array}{l} w \\ x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 0 \\ 0 \\ 0 \\ 0 \end{array}\right]\) Equation Transcription: [] [] [ ]=[ ] Text Transcription: [ 3 & -1 & 2 \\ 4 & 3 & 7 \\ -2 & 1 & 5 ] x_1 \\ x_2 \\ x_3 ]=[ 2 \\ -1 \\ 4 ] [ 3 & -2 & 0 & 1 \\ 5 & 0 & 2 & -2 \\ 3 & 1 & 4 & 7 \\ -2 & 5 & 1 & 6] [ w \\ x \\ y \\ z ]=[ 0 \\ 0 \\ 0 \\ 0 ]

In Exercises 15-16, find all values of \(k\), if any, that satisfy the equation. \(\left[\begin{array}{lll} k & 1 & 1 \end{array}\right]\left[\begin{array}{lll} 1 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & 1 & 3 \end{array}\right]\left[\begin{array}{l} k \\ 1 \\ 1 \end{array}\right]=0\) Equation Transcription: [][ ]= Text Transcription: k [ k & 1 & 1 ] [ 1 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & 1 & 3 ] [ k \\ 1 \\ 1 ]=0

In Exercises 15-16, find all values of \(k\), if any, that satisfy the equation. \(\left[\begin{array}{lll} 2 & 2 & k \end{array}\right]\left[\begin{array}{lll} 1 & 2 & 0 \\ 2 & 0 & 3 \\ 0 & 3 & 1 \end{array}\right]\left[\begin{array}{l} 2 \\ 2 \\ k \end{array}\right]=0\) Equation Transcription: [][ ]= Text Transcription: k [ 2 & 2 & k ] [ 1 & 2 & 0 \\ 2 & 0 & 3 \\ 0 & 3 & 1 ] [ 2 \\ 2 \\ k ]=0

In Exercises 17-20, use the column-row expansion of \(A B\) to express this product as a sum of matrix products. \(A=\left[\begin{array}{ll} 4 & -3 \\ 2 & -1 \end{array}\right]\) \(B=\left[\begin{array}{rrr} 0 & 1 & 2 \\ -2 & 3 & 1 \end{array}\right]\) Equation Transcription: [] [] Text Transcription: A=[ 4 & -3 \\ 2 & -1 ] B=[ 0 & 1 & 2 \\ -2 & 3 & 1 ] AB

In Exercises 17-20, use the column-row expansion of \(A B\) to express this product as a sum of matrix products. \(A=\left[\begin{array}{ll} 0 & -2 \\ 4 & -3 \end{array}\right]\) \(B=\left[\begin{array}{rrr} 1 & 4 & 1 \\ -3 & 0 & 2 \end{array}\right]\) Equation Transcription: [] [] Text Transcription: AB A=[ 0 & -2 \\ 4 & -3 ] B=[ 1 & 4 & 1 \\ -3 & 0 & 2 ]

In Exercises 17-20, use the column-row expansion of \(A B\) to express this product as a sum of matrix products. \(A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]\) \(B=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array}\right]\) Equation Transcription: [] [ ] Text Transcription: AB A=[1 & 2 & 3 \\ 4 & 5 & 6 ] B=[ 1 & 2 \\ 3 & 4 \\ 5 & 6 ]

In Exercises 17-20, use the column-row expansion of \(A B\) to express this product as a sum of matrix products. \(A=\left[\begin{array}{rrr} 0 & 4 & 2 \\ 1 & -2 & 5 \end{array}\right]\) \(B=\left[\begin{array}{rr} 2 & -1 \\ 4 & 0 \\ 1 & -1 \end{array}\right]\) Equation Transcription: [] [ ] Text Transcription: AB A=[ 0 & 4 & 2 \\ 1 & -2 & 5 ] B=[ 2 & -1 \\ 4 & 0 \\ 1 & -1 ]

For the linear system in Example 5 of Section , express the general solution that we obtained in that example as a linear combination of column vectors that contain only numerical entries. [Suggestion: Rewrite the general solution as a single column vector, then write that column vector as a sum of column vectors each of which contains at most one parameter, and then factor out the parameters.

Follow the directions of Exercise 21 for the linear system in Example 6 of Section 1.2.

In Exercises 23-24, solve the matrix equation for \(a, b, c, \text { and } d\). \(\left[\begin{array}{cc} a & 3 \\ -1 & a+b \end{array}\right]=\left[\begin{array}{cc} 4 & d-2 c \\ d+2 c & -2 \end{array}\right]\) Equation Transcription: [] [] Text Transcription: a,b,c, and d [ a & 3 \\ -1 & a+b ]=[ 4 & d-2 c \\ d+2 c & -2 ]]

In Exercises 23-24, solve the matrix equation for \(a, b, c, \text { and } d\). \(\left[\begin{array}{rr} a-b & b+a \\ 3 d+c & 2 d-c \end{array}\right]=\left[\begin{array}{ll} 8 & 1 \\ 7 & 6 \end{array}\right]\) Equation Transcription: [] [] Text Transcription: a,b,c, and d [ a-b & b+a \\ 3 d+c & 2 d-c ]=[ 8 & 1 \\ 7 & 6 ]

a. Show that if \(A\) has a row of zeros and \(B\) is any matrix for which \(AB\) is defined, then \(AB\) also has a row of zeros. b. Find a similar result involving a column of zeros. Equation Transcription: Text Transcription: A B AB AB

In each part, find a \(6 \times 6\) matrix \(\left[a_{i j}\right]\) that satisfies the stated condition. Make your answers as general as possible by using letters rather than specific numbers for the nonzero entries. a. \(a_{i j}=0 \text { if } \quad i \neq j\) b. \(a_{i j}=0 \text { if } i>j\) c. \(a_{l j}=0 \text { if } \quad i<j\) d. \(a_{i j}=0 \text { if }|i-j|>1\) Equation Transcription: Text Transcription: 6 \times 6 [a_i j] a_i j=0 \ if \quad i \neq j a_i j=0 if i>j a_l j=0 if \quad i<j a_i j=0 if |i-j|>1

In Exercises 27-28, how many \(3 \times 3\) matrices A can you find for which the equation is satisfied for all choices of \(x, y, \text { and } z\) ? \(A\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} x+y \\ x-y \\ 0 \end{array}\right]\) Equation Transcription: [ ][] Text Transcription: 3 x 3 x,y and z A[ x \\ y \\ z ]=[ x+y \\ x-y \\ 0 ]

In Exercises 27-28, how many \(3 \times 3\) matrices A can you find for which the equation is satisfied for all choices of \(x, y, \text { and } z\) ? \(A\left[\begin{array}{l} x \\ y \\ Z \end{array}\right]=\left[\begin{array}{c} x y \\ 0 \\ 0 \end{array}\right]\) Equation Transcription: [ ][] Text Transcription: 3 x 3 x,y and z A [ x \\ y \\ Z ]=[ x y \\ 0 \\ 0 ]

A matrix \(B\) is said to be a square root of a matrix \(A\) if \(B B=A\). a. Find two square roots of \(A=\left[\begin{array}{ll} 2 & 2 \\ 2 & 2 \end{array}\right]\) b. How many different square roots can you find of \(A=\left[\begin{array}{ll} 5 & 0 \\ 0 & 9 \end{array}\right]\)? c. Do you think that every \(2 \times 2\) matrix has at least one square root? Explain your reasoning. Equation Transcription: [] [] Text Transcription: B A BB=A A=[ 2 2 2 2 ] A=[ 0 9 5 0 ] 2 x 2

Let 0 denote a \(2 \times 2\) matrix, each of whose entries is zero. a. Is there a \(2 \times 2\) matrix \(A\) such that \(A \neq 0\) and \(A A=0\) ? Justify your answer. b. Is there a \(2 \times 2\) matrix \(A\) such that \(A \neq 0\) and \(A A=A\)? Justify your answer. Equation Transcription: Text Transcription: 2 x 2 2 x 2 A A \neq 0 AA=0 2 x 2 A A \neq 0 AA=A

Establish Formula (11) by using Formula (5) to show that \((A B)_{i j}=\left(c_{1} r_{1}+c_{2} r_{2}+\cdots+c_{r} r_{r}\right)_{i j}\) Equation Transcription: Text Transcription: (AB)_ij=(c_1 r_1+c_2 r_2 +. . .+c_r r_r)_ij

Find a \(4 \times 4\) matrix \(A=\left[a_{i j}\right]\) whose entries satisfy the stated condition. a. \(a_{i j}=i+j\) b. \(a_{i j}=\mathrm{i}^{\mathrm{j}-1}\) c. \(a_{i j}=\left\{\begin{array}{rll} 1 & \text { if } & |i-j|>1 \\ -1 & \text { if } & |i-j| \leq 1 \end{array}\right.\) Equation Transcription: Text Transcription: 4 x 4 A=[a_ij] a_ij=i+j a_ij = i ^ j-1 a_ij ={ -1 if |i-j| \leq1 1 if |i-j|>1

Suppose that type I items cost \(\$ 1\) each, type II items cost \(\$ 2\) each, and type III items cost \(\$ 3\) each. Also, suppose that the accompanying table describes the number of items of each type purchased during the first four months of the year. What information is represented by the following product? \(\left[\begin{array}{lll} 3 & 4 & 3 \\ 5 & 6 & 0 \\ 2 & 9 & 4 \\ 1 & 1 & 7 \end{array}\right]\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right]\) Equation Transcription: [ ][ ] Text Transcription: $1 $2 $3 [ 3 & 4 & 3 \\ 5 & 6 & 0 \\ 2 & 9 & 4 \\ 1 & 1 & 7 ][1 \\ 2 \\ 3 ]

The accompanying table shows a record of May and June unit sales for a clothing store. Let \(M\) denote the \(4 \times 3\) matrix of May sales and \(J\) the \(4 \times 3\) matrix of June sales. a. What does the matrix \(M+J\) represent? b. What does the matrix \(M-J\) represent? c. Find a column vector x for which \(M x\) provides a list of the number of shirts, jeans, suits, and raincoats sold in May. d. Find a row vector y for which \(y M\) provides a list of the number of small, medium, and large items sold in May. e. Using the matrices \(\mathrm{x} \text { and } \mathrm{y}\) that you found in parts (c) and (d), what does \(\mathrm{y} M \mathrm{x}\) represent? Equation Transcription: Text Transcription: M 4 x 3 J 4 x 3 M + J M - J Mx Mx yM x and y yMx

Prove: If \(A \text { and } B\) are \(\mathrm{n} \times n\) matrices, then \(\operatorname{tr}(A+B)=\operatorname{tr}(A)+\operatorname{tr}(B)\) Equation Transcription: Text Transcription: A and B n x n tr(A+B)=tr(A)+ tr(B)

a. Prove: If \(A B \text { and } B A\) are both defined, then \(A B \text { and } B A\) are square matrices. b. Prove: If \(A\) is an \(m \times n\) matrix and \(A(B A)\) is defined, then \(B\) is an \(\mathrm{n} \times \mathrm{m}\) matrix. Equation Transcription: Text Transcription: AB and BA AB and BA A m x n A(BA) B n x m

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. The matrix \(\left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]\) has no main diagonal. Equation Transcription: [] Text Transcription: [1 & 2 & 3 \\ 4 & 5 & 6 ]

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. b. An \(m \times n\) matrix has m column vectors and n row vectors. Equation Transcription: Text Transcription: m x n

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. c. If \(A \text { and } B\) are \(2 \times 2\) matrices, then \(A B=B A\). Equation Transcription: Text Transcription: A and B 2 x 2 AB = BA

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. d. The ith row vector of a matrix product \(A B\) can be computed by multiplying \(A\) by the ith row vector of \(B\). Equation Transcription: Text Transcription: AB A B

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. e. For every matrix \(A\), it is true that \(\left(A^{T}\right)^{T}=A\). Equation Transcription: Text Transcription: A (A^T)^T=A

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. f. If \(A \text { and } B\) are square matrices of the same order, then \(\operatorname{tr}(A B)=\operatorname{tr}(A) \operatorname{tr}(B)\) Equation Transcription: Text Transcription: A and B tr(AB)=tr(A)tr(B)

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. g. If \(A \text { and } B\) are square matrices of the same order, then \((A B)^{T}=A^{T} B^{T}\) Equation Transcription: Text Transcription: A and B (AB)^T=A^TB^T

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. h. For every square matrix \(A\), it is true that \(\operatorname{tr}\left(A^{T}\right)=\operatorname{tr}(A)\). Equation Transcription: Text Transcription: A tr(A^T)=tr(A)

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. i. If \(A\) is a \(6 \times 4\) matrix and \(B\) is an \(m \times n\) matrix such that \(B^{T} A^{T}\) is a \(2 \times 6\) matrix, then \(\mathrm{m}=4 \text { and } \mathrm{n}=2\). Equation Transcription: Text Transcription: A 6 x 4 B m x n B^TA^T 2 x 6 m=4 and n=2

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. j. If \(A\) is an \(\mathrm{n} \times \mathrm{n}\) matrix and \(c\) is a scalar, then \(\operatorname{tr}(c A)=c \operatorname{tr}(A)\). Equation Transcription: Text Transcription: A n x n c tr(cA)=c tr(A)

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. k. If \(A, B, \text { and } C\) are matrices of the same size such that \(A-C=B-C\), then \(A=B\). Equation Transcription: Text Transcription: A,B, and C A-C=B-C A=B

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. l. If \(A, B, \text { and } C\) are square matrices of the same order such that \(A C=B C\), then \(A=B\). Equation Transcription: Text Transcription: A,B, and C AC=BC A=B

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. m. If \(A B+B A\) is defined, then \(A \text { and } B\) are square matrices of the same size. Equation Transcription: Text Transcription: AB + BA A and B

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. n. If \(B\) has a column of zeros, then so does \(A B\) if this product is defined. Equation Transcription: Text Transcription: B AB

In parts (a)–(o) determine whether the statement is true or false, and justify your answer. o. If \(B\) has a column of zeros, then so does \(B A\) if this product is defined. Equation Transcription: Text Transcription: B BA

a. Compute the product \(A B\) of the matrices in Example 5, and compare your answer to that in the text. b. Use your technology utility to extract the columns of \(A\) and the rows of \(B\), and then calculate the product \(A B\) by a column-row expansion. Equation Transcription: Text Transcription: AB A B AB

Suppose that a manufacturer uses Type I items at \(\$ 1.35\) each, Type II items at \(\$ 2.15\) each, and Type III items at \(\$ 3.95\) each. Suppose also that the accompanying table describes the purchases of those items (in thousands of units) for the first quarter of the year. Find a matrix product, the computation of which produces a matrix that lists the manufacturer’s expenditure in each month of the first quarter. Compute that product. Equation Transcription: Text Transcription: $1.35 $2.15 $3.95