Let A and B be matrices such that the product AB is defined. Show that if A has two

Equality of Matrices In Exercises 14, find x and y. [ x 7 2 y] = [ 4 7 2 22]

Equality of Matrices In Exercises 14, find x and y. [ 5 y x 8] = [ 5 12 13 8]

Equality of Matrices In Exercises 14, find x and y. [ 16 3 0 4 13 2 5 15 4 4 6 0 ] = [ 16 3 0 4 13 2 2x + 1 15 3y 5 4 3x 0 ]

Equality of Matrices In Exercises 14, find x and y. [ x + 2 1 7 8 2y 2 3 2x y + 2 ] = [ 2x + 6 1 7 8 18 2 3 8 11]

Operations with Matrices In Exercises 510, find, if possible, (a) A + B, (b) A B, (c) 2A, (d) 2A B, and (e) B + 1 2A. A = [ 1 2 2 1], B = [ 3 4 2 2]

Operations with Matrices In Exercises 510, find, if possible, (a) A + B, (b) A B, (c) 2A, (d) 2A B, and (e) B + 1 2A. A = [ 6 2 3 1 4 5 ] , B = [ 1 1 1 4 5 10]

Operations with Matrices In Exercises 510, find, if possible, (a) A + B, (b) A B, (c) 2A, (d) 2A B, and (e) B + 1 2A. A = [ 2 1 1 1 1 4], B = [ 2 3 3 1 4 2]

Operations with Matrices In Exercises 510, find, if possible, (a) A + B, (b) A B, (c) 2A, (d) 2A B, and (e) B + 1 2A. A = [ 3 2 0 2 4 1 1 5 2 ] , B = [ 0 5 2 2 4 1 1 2 0 ]

Operations with Matrices In Exercises 510, find, if possible, (a) A + B, (b) A B, (c) 2A, (d) 2A B, and (e) B + 1 2A. A = [ 6 1 0 4 3 0], B = [ 8 4 1 3]

Operations with Matrices In Exercises 510, find, if possible, (a) A + B, (b) A B, (c) 2A, (d) 2A B, and (e) B + 1 2A. A = [ 3 2 1 ] , B = [462]

Find (a) c21 and (b) c13, where C = 2A 3B, A = [ 5 3 4 1 4 2], and B = [ 1 0 2 5 7 1].

Find (a) c23 and (b) c32, where C = 5A + 2B, A = [ 4 0 3 11 3 1 9 2 1 ] , and B = [ 1 4 6 0 6 4 5 11 9 ] .

Solve for x, y, and z in the matrix equation 4[ x z y 1] = 2[ y x z 1] + 2[ 4 5 x x].

Solve for x, y, z, and w in the matrix equation [ w y x x] = [ 4 2 3 1] + 2[ y z w x].

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [ 1 4 2 2], B = [ 2 1 1 8]

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [ 2 1 2 4], B = [ 4 2 1 2]

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [ 2 5 2 1 1 2 3 2 3 ] , B = [ 0 4 4 1 1 1 2 3 2 ]

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [ 1 2 3 1 1 1 7 8 1 ] , B = [ 1 2 1 1 1 3 2 1 2 ]

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [ 2 3 1 1 4 6 ] , B = [ 0 4 8 1 0 1 0 2 7 ]

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [ 3 3 4 2 0 2 1 4 4 ] , B = [ 1 2 1 2 1 2 ]

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [321], B = [ 2 3 0 ]

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [ 1 2 2 1 ] , B = [2132]

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [ 1 4 0 3 5 2 ] , B = [ 1 0 2 7]

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [ 2 5 3 2], B = [ 2 1 2 1 3 1 ]

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [ 0 4 8 1 0 1 0 2 7 ] , B = [ 2 3 1 ]

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [ 2 3 2 1 1 1 2 2 2 ] , B = [ 4 1 2 0 2 1 1 3 4 3 1 3 ]

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [ 6 2 1 6 ] , B = [10 12]

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [ 1 6 0 13 3 8 2 17 4 20], B = [ 1 4 6 2]

Matrix Size In Exercises 2936, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 4 B: 3 4 C: 4 2 D: 4 2 E: 4 3 If defined, determine the size of the matrix. If not defined, explain why. A + B

Matrix Size In Exercises 2936, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 4 B: 3 4 C: 4 2 D: 4 2 E: 4 3 If defined, determine the size of the matrix. If not defined, explain why. C + E

Matrix Size In Exercises 2936, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 4 B: 3 4 C: 4 2 D: 4 2 E: 4 3 If defined, determine the size of the matrix. If not defined, explain why. 1 2D

Matrix Size In Exercises 2936, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 4 B: 3 4 C: 4 2 D: 4 2 E: 4 3 If defined, determine the size of the matrix. If not defined, explain why. 4A

Matrix Size In Exercises 2936, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 4 B: 3 4 C: 4 2 D: 4 2 E: 4 3 If defined, determine the size of the matrix. If not defined, explain why. AC

Matrix Size In Exercises 2936, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 4 B: 3 4 C: 4 2 D: 4 2 E: 4 3 If defined, determine the size of the matrix. If not defined, explain why. BE

Matrix Size In Exercises 2936, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 4 B: 3 4 C: 4 2 D: 4 2 E: 4 3 If defined, determine the size of the matrix. If not defined, explain why. E 2A

Matrix Size In Exercises 2936, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 4 B: 3 4 C: 4 2 D: 4 2 E: 4 3 If defined, determine the size of the matrix. If not defined, explain why. 2D + C

Solving a Matrix Equation In Exercises 37 and 38, solve the matrix equation Ax = 0. A = [ 2 1 1 2 1 2], x = [ x1 x2 x3 ] , 0 = [ 0 0]

Solving a Matrix Equation In Exercises 37 and 38, solve the matrix equation Ax = 0. A = [ 1 1 0 2 1 1 1 0 1 3 1 2 ] , x = [ x1 x2 x3 x4 ] , 0 = [ 0 0 0 ]

Solving a System of Linear Equations In Exercises 3948, write the system of linear equations in the form Ax = b and solve this matrix equation for x. x1 + x2 = 4 2x1 + x2 = 0

Solving a System of Linear Equations In Exercises 3948, write the system of linear equations in the form Ax = b and solve this matrix equation for x. 2x1 + 3x2 = 5 x1 + 4x2 = 10

Solving a System of Linear Equations In Exercises 3948, write the system of linear equations in the form Ax = b and solve this matrix equation for x. 2x1 3x2 = 4 6x1 + x2 = 36

Solving a System of Linear Equations In Exercises 3948, write the system of linear equations in the form Ax = b and solve this matrix equation for x. 4x1 + 9x2 = 13 x1 3x2 = 12

Solving a System of Linear Equations In Exercises 3948, write the system of linear equations in the form Ax = b and solve this matrix equation for x. x1 2x2 + 3x3 = 9 x1 + 3x2 x3 = 6 2x1 5x2 + 5x3 = 17

Solving a System of Linear Equations In Exercises 3948, write the system of linear equations in the form Ax = b and solve this matrix equation for x. x1 + x2 3x3 = 1 x1 + 2x2 = 1 x1 x2 + x3 = 2

Solving a System of Linear Equations In Exercises 3948, write the system of linear equations in the form Ax = b and solve this matrix equation for x. x1 5 x2 + 2x3 = 20 3x1 + x2 x3 = 8 2x2 + 5x3 = 16

Solving a System of Linear Equations In Exercises 3948, write the system of linear equations in the form Ax = b and solve this matrix equation for x. x1 x2 + 4x3 = 17 x1 + 3x2 = 11 6x2 + 5x3 = 40

Solving a System of Linear Equations In Exercises 3948, write the system of linear equations in the form Ax = b and solve this matrix equation for x. 2x1 x2 + x4 = 3 3x2 x3 x4 = 3 x1 + x3 3x4 = 4 x1 + x2 + 2x3 =

Solving a System of Linear Equations In Exercises 3948, write the system of linear equations in the form Ax = b and solve this matrix equation for x. x1 + x2 = 0 x2 + x3 = 0 x3 + x4 = 0 x4 + x5 = 0 x1 + x2 x3 + x4 x5 = 5

Writing a Linear Combination In Exercises 4952, write the column matrix b as a linear combination of the columns of A. A = [ 1 3 1 3 2 1], b = [ 1 7]

Writing a Linear Combination In Exercises 4952, write the column matrix b as a linear combination of the columns of A. A = [ 1 1 0 2 0 1 4 2 3 ] , b = [ 1 3 2 ]

Writing a Linear Combination In Exercises 4952, write the column matrix b as a linear combination of the columns of A. A = [ 1 1 2 1 0 1 5 1 1 ] , b = [ 3 1 0 ]

Writing a Linear Combination In Exercises 4952, write the column matrix b as a linear combination of the columns of A. A = [ 3 3 4 5 4 8 ] , b = [ 22 4 32]

Solving a Matrix Equation In Exercises 53 and 54, solve for A. [ 1 3 2 5] A = [ 1 0 0 1]

Solving a Matrix Equation In Exercises 53 and 54, solve for A. [ 2 3 1 2] A = [ 1 0 0 1]

Solving a Matrix Equation In Exercises 55 and 56, solve the matrix equation for a, b, c, and d. [ 1 3 2 4] [ a c b d] = [ 6 19 3 2]

Solving a Matrix Equation In Exercises 55 and 56, solve the matrix equation for a, b, c, and d. [ a c b d] [ 2 3 1 1] = [ 3 4 17 1]

Diagonal Matrix In Exercises 57 and 58, find the product AA for the diagonal matrix. A square matrix is a diagonal matrix when all entries that are not on the main diagonal are zero. A = [ 1 0 0 0 2 0 0 0 3 ]

Diagonal Matrix In Exercises 57 and 58, find the product AA for the diagonal matrix. A square matrix is a diagonal matrix when all entries that are not on the main diagonal are zero. A = [ 2 0 0 0 3 0 0 0 0 ]

Finding Products of Diagonal Matrices In Exercises 59 and 60, find the products AB and BA for the diagonal matrices. A = [ 2 0 0 3], B = [ 5 0 0 4]

Finding Products of Diagonal Matrices In Exercises 59 and 60, find the products AB and BA for the diagonal matrices. A = [ 3 0 0 0 5 0 0 0 0 ] , B = [ 7 0 0 0 4 0 0 0 12]

Guided Proof Prove that if A and B are diagonal matrices (of the same size), then AB = BA. Getting Started: To prove that the matrices AB and BA are equal, you need to show that their corresponding entries are equal. (i) Begin your proof by letting A = [aij] and B = [bij] be two diagonal n n matrices. (ii) The ijth entry of the product AB is cij = n k=1 aikbkj. (iii) Evaluate the entries cij for the two cases i j and i = j. (iv) Repeat this analysis for the product BA

Writing Let A and B be 3 3 matrices, where A is diagonal. (a) Describe the product AB. Illustrate your answer with examples. (b) Describe the product BA. Illustrate your answer with examples. (c) How do the results in parts (a) and (b) change when the diagonal entries of A are all equal?

Trace of a Matrix In Exercises 6366, find the trace of the matrix. The trace of an n n matrix A is the sum of the main diagonal entries. That is, Tr(A) = a11 + a22 + . . . + ann. [ 1 0 3 2 2 1 3 4 3 ]

Trace of a Matrix In Exercises 6366, find the trace of the matrix. The trace of an n n matrix A is the sum of the main diagonal entries. That is, Tr(A) = a11 + a22 + . . . + ann. [ 1 0 0 0 1 0 0 0 1 ]

Trace of a Matrix In Exercises 6366, find the trace of the matrix. The trace of an n n matrix A is the sum of the main diagonal entries. That is, Tr(A) = a11 + a22 + . . . + ann. [ 1 0 4 0 0 1 2 0 2 1 1 5 1 2 0 1 ]

Trace of a Matrix In Exercises 6366, find the trace of the matrix. The trace of an n n matrix A is the sum of the main diagonal entries. That is, Tr(A) = a11 + a22 + . . . + ann. [ 1 4 3 2 4 0 6 1 3 6 2 1 2 1 1 3 ]

Proof Prove that each statement is true when A and B are square matrices of order n and c is a scalar. (a) Tr(A + B) = Tr(A) + Tr(B) (b) Tr(cA) = cTr(A)

Proof Prove that if A and B are square matrices of order n, then Tr(AB) = Tr(BA).

Find conditions on w, x, y, and z such that AB = BA for the matrices below. A = [ w y x z] and B = [ 1 1 1 1]

Verify AB = BA for the matrices below. A = [ cos sin sin cos ] and B = [ cos sin sin cos ]

Show that the matrix equation has no solution. [ 1 1 1 1] A = [ 1 0 0 1]

Show that no 2 2 matrices A and B exist that satisfy the matrix equation AB BA = [ 1 0 0 1].

Exploration Let i = 1 and let A = [ i 0 0 i ] and B = [ 0 i i 0]. (a) Find A2, A3, and A4. (Note: A2 = AA, A3 = AAA = A2A, and so on.) Identify any similarities with i 2, i 3, and i 4. (b) Find and identify B2.

Guided Proof Prove that if the product AB is a square matrix, then the product BA is defined. Getting Started: To prove that the product BA is defined, you need to show that the number of columns of B equals the number of rows of A. (i) Begin your proof by noting that the number of columns of A equals the number of rows of B. (ii) Then assume that A has size m n and B has size n p. (iii) Use the hypothesis that the product AB is a square matrix.

Proof Prove that if both products AB and BA are defined, then AB and BA are square matrices.

Let A and B be matrices such that the product AB is defined. Show that if A has two identical rows, then the corresponding two rows of AB are also identical.

Let A and B be n n matrices. Show that if the ith row of A has all zero entries, then the ith row of AB will have all zero entries. Give an example using 2 2 matrices to show that the converse is not true.

CAPSTONE Let matrices A and B be of sizes 3 2 and 2 2, respectively. Answer each question and explain your answers. (a) Is it possible that A = B? (b) Is A + B defined? (c) Is AB defined? If so, is it possible that AB = BA?

Agriculture A fruit grower raises two crops, apples and peaches. The grower ships each of these crops to three different outlets. In the matrix A = [ 125 100 100 175 75 125] aij represents the number of units of crop i that the grower ships to outlet j. The matrix B = [$3.50 $6.00] represents the profit per unit. Find the product BA and state what each entry of the matrix represents.

Manufacturing A corporation has three factories, each of which manufactures acoustic guitars and electric guitars. In the matrix A = [ 70 35 50 100 25 70] aij represents the number of guitars of type i produced at factory j in one day. Find the production levels when production increases by 20%.

Politics In the matrix each entry pij (i j) represents the proportion of the voting population that changes from party j to party i, and pii represents the proportion that remains loyal to party i from one election to the next. Find and interpret the product of P with itself.

Population The matrices show the numbers of people (in thousands) who lived in each region of the United States in 2010 and 2013. The regional populations are separated into three age categories. (Source: U.S. Census Bureau) (a) The total population in 2010 was approximately 309 million and the total population in 2013 was about 316 million. Rewrite the matrices to give the information as percents of the total population. (b) Write a matrix that gives the changes in the percents of the population in each region and age group from 2010 to 2013. (c) Based on the result of part (b), which age group(s) show relative growth from 2010 to 2013?

Block Multiplication In Exercises 83 and 84, perform the block multiplication of matrices A and B. If matrices A and B are each partitioned into four submatrices A = [ A11 A21 A12 A22] and B = [ B11 B21 B12 B22] then you can block multiply A and B, provided the sizes of the submatrices are such that the matrix multiplications and additions are defined. AB = [ A11 A21 A12 A22] [ B11 B21 B12 B22] = [ A11B11 + A12B21 A21B11 + A22B21 A11B12 + A12B22 A21B12 + A22B22] A = [ 1 0 0 2 1 0 0 0 2 0 0 1 ] , B = [ 1 1 0 0 2 1 0 0 0 0 1 3 ]

Block Multiplication In Exercises 83 and 84, perform the block multiplication of matrices A and B. If matrices A and B are each partitioned into four submatrices A = [ A11 A21 A12 A22] and B = [ B11 B21 B12 B22] then you can block multiply A and B, provided the sizes of the submatrices are such that the matrix multiplications and additions are defined. AB = [ A11 A21 A12 A22] [ B11 B21 B12 B22] = [ A11B11 + A12B21 A21B11 + A22B21 A11B12 + A12B22 A21B12 + A22B22] A = [ 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 ] , B = [ 1 5 1 5 2 6 2 6 3 7 3 7 4 8 4 8 ]

True or False? In Exercises 85 and 86, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) For the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix. (b) The system Ax = b is consistent if and only if b can be expressed as a linear combination of the columns of A, where the coefficients of the linear combination are a solution of the system.

True or False? In Exercises 85 and 86, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) If A is an m n matrix and B is an n r matrix, then the product AB is an m r matrix. (b) The matrix equation Ax = b, where A is the coefficient matrix and x and b are column matrices, can be used to represent a system of linear equations.

Evaluating an Expression In Exercises 16, evaluate the expression

In Exercises 1–4, find x and y. \(\left[\begin{array}{rr} x & -2 \\ 7 & y \end{array}\right]=\left[\begin{array}{rr} -4 & -2 \\ 7 & 22 \end{array}\right] \) Text Transcription: [_7^x _y^-2] = [_7^-4 _22^-2]

In Exercises 1–4, find x and y. \(\left[\begin{array}{rr} -5 & x \\ y & 8 \end{array}\right]=\left[\begin{array}{rr} -5 & 13 \\ 12 & 8 \end{array}\right] \) Text Transcription: [_y^-5 _8^x] = [_12^-5 _8^13]

In Exercises 1–4, find x and y. \(\left[\begin{array}{rrrr} 16 & 4 & 5 & 4 \\ -3 & 13 & 15 & 6 \\ 0 & 2 & 4 & 0 \end{array}\right]=\left[\begin{array}{rrrr} 16 & 4 & 2 x+1 & 4 \\ -3 & 13 & 15 & 3 x \\ 0 & 2 & 3 y-5 & 0 \end{array}\right] \) Text Transcription: [_0^-3^16 _2^13^4 _4^15^5 _0^6^4] = [_0^-3^16 _2^13^4 _3y-5^15^2x+1 _0^3x^4]

In Exercises 1–4, find x and y. \(\left[\begin{array}{rrr} x+2 & 8 & -3 \\ 1 & 2 y & 2 x \\ 7 & -2 & y+2 \end{array}\right]=\left[\begin{array}{rrr} 2 x+6 & 8 & -3 \\ 1 & 18 & -8 \\ 7 & -2 & 11 \end{array}\right] \) Text Transcription: [_7^1^x+2 _-2^2y^8 _y+2^2x^-3] = [_7^1^2x+6 _-2^18^8 _11^-8-3]

In Exercises 5–10, find, if possible, (a) A B, (b) A B, (c) 2A, (d) 2A B, and \(B+\frac{1}{2} A\). \(A=\left[\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right], \quad B=\left[\begin{array}{rr} -3 & -2 \\ 4 & 2 \end{array}\right] \) Text Transcription: A = [_2^1 _1^2], B = [_4^-3 _2^-2]

In Exercises 5–10, find, if possible, (a) A B, (b) A B, (c) 2A, (d) 2A B, and \(B+\frac{1}{2} A\). \(A=\left[\begin{array}{rr} 6 & -1 \\ 2 & 4 \\ -3 & 5 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 4 \\ -1 & 5 \\ 1 & 10 \end{array}\right] \) Text Transcription: A = [_-3^2^6 _5^4^-1], B = [_1^-1^1 _10^5^4]

In Exercises 5–10, find, if possible, (a) A B, (b) A B, (c) 2A, (d) 2A B, and \(B+\frac{1}{2} A\). \(A=\left[\begin{array}{rrr} 2 & 1 & 1 \\ -1 & -1 & 4 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & -3 & 4 \\ -3 & 1 & -2 \end{array}\right] \) Text Transcription: A = [_-1^2 _-1^1 _4^1], B = [_-3^2 _1^-3 _-2^4]

In Exercises 5–10, find, if possible, (a) A B, (b) A B, (c) 2A, (d) 2A B, and \(B+\frac{1}{2} A\). \(A=\left[\begin{array}{rrr} 3 & 2 & -1 \\ 2 & 4 & 5 \\ 0 & 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{lll} 0 & 2 & 1 \\ 5 & 4 & 2 \\ 2 & 1 & 0 \end{array}\right] \) Text Transcription: A = [_0^2^3 _1^4^2 _2^5^-1], B = [_2^5^0 _1^4^2 _0^2^1]

In Exercises 5–10, find, if possible, (a) A B, (b) A B, (c) 2A, (d) 2A B, and \(B+\frac{1}{2} A\). \(A=\left[\begin{array}{rrr} 6 & 0 & 3 \\ -1 & -4 & 0 \end{array}\right], \quad B=\left[\begin{array}{ll} 8 & -1 \\ 4 & -3 \end{array}\right] \) Text Transcription: A = [_-1^6 _-4^0 _0^3], B = [_4^8 _-3^-1]

In Exercises 5–10, find, if possible, (a) A B, (b) A B, (c) 2A, (d) 2A B, and \(B+\frac{1}{2} A\). \(A=\left[\begin{array}{r} 3 \\ 2 \\ -1 \end{array}\right], \quad B=\left[\begin{array}{lll} -4 & 6 & 2 \end{array}\right] \) Text Transcription: A = [_-1^2^3], B = [-4 6 2]

Find (a) \(c_{21}\) and (b) \(c_{13}\), where C = 2A ? 3B, \(A=\left[\begin{array}{rrr} 5 & 4 & 4 \\ -3 & 1 & 2 \end{array}\right] \), and \(B=\left[\begin{array}{rrr} 1 & 2 & -7 \\ 0 & -5 & 1 \end{array}\right] \). Text Transcription: c_21 c_13 A = [_-3^5 _1^4 _2^4] B = [_0^1 _-5^2 _1^-7]

Find (a) \(c_{23}\) and (b) \(c_{32}\), where C = 5A + 2B, \(A=\left[\begin{array}{rrr} 4 & 11 & -9 \\ 0 & 3 & 2 \\ -3 & 1 & 1 \end{array}\right]\), and \(B=\left[\begin{array}{rrr} 1 & 0 & 5 \\ -4 & 6 & 11 \\ -6 & 4 & 9 \end{array}\right] \). Text Transcription: c_23 c_32 A = [_-3^0^4 _1^3^11 _1^2^-9] B = [_-6^-4^1 _4^6^0 _0^11^5]

Solve for x, y, and z in the matrix equation \(4\left[\begin{array}{rr} x & y \\ z & -1 \end{array}\right]=2\left[\begin{array}{rr} y & z \\ -x & 1 \end{array}\right]+2\left[\begin{array}{rr} 4 & x \\ 5 & -x \end{array}\right] \). Text Transcription: 4[_z^x _-1^y] = 2[_-x^y _1^z] + 2[_5^4 _-x^x]

Solve for x, y, z, and w in the matrix equation \(\left[\begin{array}{ll} w & x \\ y & x \end{array}\right]=\left[\begin{array}{rr} -4 & 3 \\ 2 & -1 \end{array}\right]+2\left[\begin{array}{ll} y & w \\ z & x \end{array}\right] \). Text Transcription: [_y^w _x^x] = [_2^-4 _-1^3] +2[_z^y _x^w]

In Exercises 15–28, find, if possible, (a) AB and (b) BA. \(A=\left[\begin{array}{ll} 1 & 2 \\ 4 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -1 \\ -1 & 8 \end{array}\right] \). Text Transcription: A = [_4^1 _2^2], B = [_-1^2 _8^-1]

In Exercises 15–28, find, if possible, (a) AB and (b) BA. \(A=\left[\begin{array}{rr} 2 & -2 \\ -1 & 4 \end{array}\right], \quad B=\left[\begin{array}{rr} 4 & 1 \\ 2 & -2 \end{array}\right] \). Text Transcription: A = [_-1^2 _4^-2], B = [_2^4 _-2^1]

In Exercises 15–28, find, if possible, (a) AB and (b) BA. \(A=\left[\begin{array}{rrr} 2 & -1 & 3 \\ 5 & 1 & -2 \\ 2 & 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rrr} 0 & 1 & 2 \\ -4 & 1 & 3 \\ -4 & -1 & -2 \end{array}\right] \). Text Transcription: A = [_2^5^2 _ 2^1^-1 _3^-2^3], B = [_-4^-4^0 _-1^1^1 _-2^3^2]

In Exercises 15–28, find, if possible, (a) AB and (b) BA. \(A=\left[\begin{array}{rrr} 1 & -1 & 7 \\ 2 & -1 & 8 \\ 3 & 1 & -1 \end{array}\right], \quad B=\left[\begin{array}{rrr} 1 & 1 & 2 \\ 2 & 1 & 1 \\ 1 & -3 & 2 \end{array}\right] \). Text Transcription: A = [_1^-3^2 _6^4^1], B = [_8^4^0 _-1^0^-1 _7^2^0]

In Exercises 15–28, find, if possible, (a) AB and (b) BA. \(A=\left[\begin{array}{rr} 2 & 1 \\ -3 & 4 \\ 1 & 6 \end{array}\right], \quad B=\left[\begin{array}{rrr} 0 & -1 & 0 \\ 4 & 0 & 2 \\ 8 & -1 & 7 \end{array}\right] \). Text Transcription: A = [_1^-3^2 _6^4^1], B = [_8^4^0 _-1^0^-1 _7^2^0]

In Exercises 15–28, find, if possible, (a) AB and (b) BA. \(A=\left[\begin{array}{rrr} 3 & 2 & 1 \\ -3 & 0 & 4 \\ 4 & -2 & -4 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 2 \\ 2 & -1 \\ 1 & -2 \end{array}\right] \). Text Transcription: A = [_4^-3^3 _-2^0^2 _-4^4^1], B = [_1^2^1 _^-2^-1^2]

In Exercises 15–28, find, if possible, (a) AB and (b) BA. \(A=\left[\begin{array}{lll} 3 & 2 & 1 \end{array}\right], \quad B=\left[\begin{array}{l} 2 \\ 3 \\ 0 \end{array}\right] \). Text Transcription: A = [3 2 1 ]. B = [_0^3^2]

In Exercises 15–28, find, if possible, (a) AB and (b) BA. \(A=\left[\begin{array}{r} -1 \\ 2 \\ -2 \\ 1 \end{array}\right], \quad B=\left[\begin{array}{llll} 2 & 1 & 3 & 2 \end{array}\right] \). Text Transcription: A = [_1^-2^2^-1], B = [2 1 3 2]

In Exercises 15–28, find, if possible, (a) AB and (b) BA. \(A=\left[\begin{array}{rr} -1 & 3 \\ 4 & -5 \\ 0 & 2 \end{array}\right], \quad B=\left[\begin{array}{ll} 1 & 2 \\ 0 & 7 \end{array}\right] \). Text Transcription: A = [_0^4^-1 _2^-5^3], B = [_0^1 _7^2]

In Exercises 15–28, find, if possible, (a) AB and (b) BA. \(A=\left[\begin{array}{rr} 2 & -3 \\ 5 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 1 \\ 1 & 3 \\ 2 & -1 \end{array}\right] \). Text Transcription: A = [_5^2 _2^-3], B = [_2^1^2 _-1^3^1]

In Exercises 15–28, find, if possible, (a) AB and (b) BA. \(A=\left[\begin{array}{rrr} 0 & -1 & 0 \\ 4 & 0 & 2 \\ 8 & -1 & 7 \end{array}\right], \quad B=\left[\begin{array}{r} 2 \\ -3 \\ 1 \end{array}\right] \). Text Transcription: A = [_8^4^0 _-1^0^-1 _7^2^0], B = [_1^-3^2]

In Exercises 15–28, find, if possible, (a) AB and (b) BA. \(A=\left[\begin{array}{rrr} 2 & 1 & 2 \\ 3 & -1 & -2 \\ -2 & 1 & -2 \end{array}\right], \quad B=\left[\begin{array}{rrrr} 4 & 0 & 1 & 3 \\ -1 & 2 & -3 & -1 \\ -2 & 1 & 4 & 3 \end{array}\right] \). Text Transcription: A = [_-2^3^2 _1^-1^1 _-2^-2^2], B = [_-2^-1^4 _1^2^0 _4^-3^1 _3^-1^3]

In Exercises 15–28, find, if possible, (a) AB and (b) BA. \(A=\left[\begin{array}{r} 6 \\ -2 \\ 1 \\ 6 \end{array}\right], \quad B=[10 \). Text Transcription: A = [_6^1^-2^6], B = [10 12]

Finding Products of Two Matrices In Exercises 1528, find, if possible, (a) AB and (b) BA. A = [ 1 6 0 13 3 8 2 17 4 20], B = [ 1 4 6 2]

In Exercises 29–36, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 x 4 B: 3 x 4 C: 4 x 2 D: 4 x2 E: 4 x 3 If defined, determine the size of the matrix. If not defined, explain why. A + B

In Exercises 29–36, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 x 4 B: 3 x 4 C: 4 x 2 D: 4 x2 E: 4 x 3 If defined, determine the size of the matrix. If not defined, explain why. C + E

Matrix Size In Exercises 2936, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 4 B: 3 4 C: 4 2 D: 4 2 E: 4 3 If defined, determine the size of the matrix. If not defined, explain why. 1 2D

In Exercises 29–36, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 x 4 B: 3 x 4 C: 4 x 2 D: 4 x2 E: 4 x 3 If defined, determine the size of the matrix. If not defined, explain why. ?4A

In Exercises 29–36, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 x 4 B: 3 x 4 C: 4 x 2 D: 4 x2 E: 4 x 3 If defined, determine the size of the matrix. If not defined, explain why. AC

In Exercises 29–36, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 x 4 B: 3 x 4 C: 4 x 2 D: 4 x2 E: 4 x 3 If defined, determine the size of the matrix. If not defined, explain why. BE

In Exercises 29–36, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 x 4 B: 3 x 4 C: 4 x 2 D: 4 x2 E: 4 x 3 If defined, determine the size of the matrix. If not defined, explain why. E ? 2A

In Exercises 29–36, let A, B, C, D, and E be matrices with the sizes shown below. A: 3 x 4 B: 3 x 4 C: 4 x 2 D: 4 x2 E: 4 x 3 If defined, determine the size of the matrix. If not defined, explain why. 2D + C

In Exercises 37 and 38, solve the matrix equation Ax 0. \(A=\left[\begin{array}{rrr} 2 & -1 & -1 \\ 1 & -2 & 2 \end{array}\right], \quad \mathbf{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right], \quad \mathbf{0}=\left[\begin{array}{l} 0 \\ 0 \end{array}\right] \) Text Transcription: A = [_1^2 _-2^-1 _2^-1], x = [_x_3^2 x_2^ x_1], 0 = [_0^0]

In Exercises 37 and 38, solve the matrix equation Ax 0. \(A=\left[\begin{array}{rrrr} 1 & 2 & 1 & 3 \\ 1 & -1 & 0 & 1 \\ 0 & 1 & -1 & 2 \end{array}\right], \quad \mathbf{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right], \quad \mathbf{0}=\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right] \) Text Transcription: A = [_0^1^1 _1^-1^2 _-1^0^1 _2^1^3], x = [_x_4^x_3^x_2^x_1], 0 = [_0^0^0]

In Exercises 39–48, write the system of linear equations in the form Ax b and solve this matrix equation for x. \(?x_{1} + x_{2} = 4\) \(?2x_{1} + x_{2} = 0\) Text Transcription: ?x_1 + x_2 = 4 ?2x_1 + x_2 = 0

In Exercises 39–48, write the system of linear equations in the form Ax b and solve this matrix equation for x. \(2x_{1} + 3x_{2} = 5\) \(x_{1} + 4x_{2} = 10\) Text Transcription: 2x_1 + 3x_2 = 5 x_1 + 4x_2 = 10

In Exercises 39–48, write the system of linear equations in the form Ax b and solve this matrix equation for x. \(?2x_{1} ? 3x_{2} = ?4\) \(6x_1 + x_2 = ?36\) Text Transcription: ?2x_1 ? 3x_2 = ?4 6x_1 + x_2 = ?36

In Exercises 39–48, write the system of linear equations in the form Ax b and solve this matrix equation for x. \(?4x_{1} + 9x_{2} = ?13\) \(x_{1} ? 3x_{2} = 12\) Text Transcription: ?4x_1 + 9x_2 = ?13 x_1 ? 3x_2 = 12

In Exercises 39–48, write the system of linear equations in the form Ax b and solve this matrix equation for x. \(x_1 ? 2x_2 + 3x_3 = 9\) \(?x_{1} + 3x_{2} ? x_{3} = ?6\) \(2x_1 ? 5x_2 + 5x_3 = 17\) Text Transcription: x_1 ? 2x_2 + 3x_3 = 9 ?x_1 + 3x_2 ? x_3 = ?6 2x_1 ? 5x_2 + 5x_3 = 17

In Exercises 39–48, write the system of linear equations in the form Ax b and solve this matrix equation for x. \(x_{1} + x_{2} ? 3x_{3} = ?1\) \(?x_{1} + 2x_{2} = 1\) \(x_1 ? x_2 + x_3 = 2\) Text Transcription: x_1 + x_2 ? 3x_3 = ?1 ?x_1 + 2x_2 = 1 x_{1} ? x_{2} + x_{3} = 2

In Exercises 39–48, write the system of linear equations in the form Ax b and solve this matrix equation for x. \(x_{1} ? 5 x_{2} + 2x_{3} = ?20\) \(?3x_{1} + x_{2} ? x_{3} = 8\) \(?2x_{2} + 5x_{3} = ?16\) Text Transcription: x_1 ? 5 x_2 + 2x_3 = ?20 ?3x_1 + x_2 ? x_3 = 8 ?2x_2 + 5x_3 = ?16

In Exercises 39–48, write the system of linear equations in the form Ax b and solve this matrix equation for x. \(x_{1} ? x_{2} + 4x_{3} = 17\) \(x_{1} + 3x_{2} = ?11\) \(?6x_{2} + 5x_{3} = 40\) Text Transcription: x_1 ? x_2 + 4x_3 = 17 x_1 + 3x_2 = ?11 ?6x_2 + 5x_3 = 40

In Exercises 39–48, write the system of linear equations in the form Ax b and solve this matrix equation for x. \(2x_{1} ? x_{2} + x4 = 3\) \(3x_2 ? x_3 ? x_4 = ?3\) \(x_{1} + x_{3} ? 3x_{4} = ?4\) \(x_{1} + x_{2} + 2x_{3} = 0\) Text Transcription: 2x_1 ? x_2 + x4 = 3 3x_{2} ? x_{3} ? x_{4} = ?3 x_1 + x_3 ? 3x_4 = ?4 x_1 + x_2 + 2x_3 = 0

In Exercises 39–48, write the system of linear equations in the form Ax b and solve this matrix equation for x. \(x_{1} + x_{2} = 0\) \(x_{2} + x_{3} = 0\) \(x_{3} + x_{4} = 0\) \(x_{4} + x_{5} = 0\) \(?x_{1} + x_{2} ? x_{3} + x_{4} ? x_{5} = 5\) Text Transcription: x_1 + x_2 = 0 x_2 + x_3 = 0 x_3 + x_4 = 0 x_4 + x_5 = 0 ?x_1 + x_2 ? x_3 + x_4 ? x_5 = 5

In Exercises 49–52, write the column matrix b as a linear combination of the columns of A. \(A=\left[\begin{array}{lll} 1 & -1 & 2 \\ 3 & -3 & 1 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{r} -1 \\ 7 \end{array}\right] \) Text Transcription: A = [_3^1 _-3^-1 _1^2], b = [_7^-1]

In Exercises 49–52, write the column matrix b as a linear combination of the columns of A. \(A=\left[\begin{array}{rrr} 1 & 2 & 4 \\ -1 & 0 & 2 \\ 0 & 1 & 3 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l} 1 \\ 3 \\ 2 \end{array}\right] \) Text Transcription: A = [_0^-1^1 _1^0^2 _3^2^4], b = [_2^3^1]

In Exercises 49–52, write the column matrix b as a linear combination of the columns of A. \(A=\left[\begin{array}{rrr} 1 & 1 & -5 \\ 1 & 0 & -1 \\ 2 & -1 & -1 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l} 3 \\ 1 \\ 0 \end{array}\right] \) Text Transcription: A = [_2^1^1 _-1^0^1 _-1^-1^-5]. B = [_0^1^3]

In Exercises 49–52, write the column matrix b as a linear combination of the columns of A. \(A=\left[\begin{array}{rr} -3 & 5 \\ 3 & 4 \\ 4 & -8 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{r} -22 \\ 4 \\ 32 \end{array}\right] \) Text Transcription: A = [_4^3^-3 _8^4^5], b = [_32^4^-22]

In Exercises 53 and 54, solve for A. \(\left[\begin{array}{ll} 1 & 2 \\ 3 & 5 \end{array}\right] A=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \) Text Transcription: [_3^1 _5^2] A = [_0^1 _1^0]

In Exercises 53 and 54, solve for A. \(\left[\begin{array}{ll} 2 & -1 \\ 3 & -2 \end{array}\right] A=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \) Text Transcription: [_3^2 _-2^-1] A = [_0^1 _1^0]

In Exercises 55 and 56, solve the matrix equation for a, b, c, and d. \(\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{rr} 6 & 3 \\ 19 & 2 \end{array}\right] \) Text Transcription: [_3^1 _4^2] [_c^a _d^b] = [_19^6 _2^3]

In Exercises 55 and 56, solve the matrix equation for a, b, c, and d. \(\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\left[\begin{array}{ll} 2 & 1 \\ 3 & 1 \end{array}\right]=\left[\begin{array}{rr} 3 & 17 \\ 4 & -1 \end{array}\right] \) Text Transcription: [_c^a _d^b] [_3^2 _1^1] = [_4^3 _-1^17]

In Exercises 57 and 58, find the product AA for the diagonal matrix. A square matrix is a diagonal matrix when all entries that are not on the main diagonal are zero. \(A=\left[\begin{array}{rrr} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array}\right] \) Text Transcription: A = [_0^0^-1 _0^2^0 _3^0^0]

In Exercises 57 and 58, find the product AA for the diagonal matrix. A square matrix is a diagonal matrix when all entries that are not on the main diagonal are zero. \(A=\left[\begin{array}{rrr} 2 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 0 \end{array}\right] \) Text Transcription: A = [_0^0^2 _0^-3^0 _0^0^0]

In Exercises 59 and 60, find the products AB and BA for the diagonal matrices. \(A=\left[\begin{array}{rr} 2 & 0 \\ 0 & -3 \end{array}\right], \quad B=\left[\begin{array}{rr} -5 & 0 \\ 0 & 4 \end{array}\right] \) Text Transcription: A = [_0^2 _-3^0], B = [_0^-5 _4^0]

In Exercises 59 and 60, find the products AB and BA for the diagonal matrices. \(A=\left[\begin{array}{rrr} 3 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} -7 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 12 \end{array}\right] \) Text Transcription: A = [_0^0^3 _0^-5^0 _0^0^0], B = [_0^0^-7 _0^4^0 _12^0^0]

Guided Proof Prove that if A and B are diagonal matrices (of the same size), then AB = BA. Getting Started: To prove that the matrices AB and BA are equal, you need to show that their corresponding entries are equal. (i) Begin your proof by letting A = [aij] and B = [bij] be two diagonal n n matrices. (ii) The ijth entry of the product AB is cij = n k=1 aikbkj. (iii) Evaluate the entries cij for the two cases i j and i = j. (iv) Repeat this analysis for the product BA

Let A and B be 3 × 3 matrices, where A is diagonal. (a) Describe the product AB. Illustrate your answer with examples. (b) Describe the product BA. Illustrate your answer with examples. (c) How do the results in parts (a) and (b) change when the diagonal entries of A are all equal?

In Exercises 63–66, find the trace of the matrix. The trace of an n n matrix A is the sum of the main diagonal entries. That is, \(Tr(A) = a_{11} + a_{22} + \cdot + a_{nn}\). \(\left[\begin{array}{rrr} 1 & 2 & 3 \\ 0 & -2 & 4 \\ 3 & 1 & 3 \end{array}\right] \) Text Transcription: Tr(A) = a_11 + a_22 + cdot + a_nn [_3^0^1 _1^-2^2 _3^4^3]

In Exercises 63–66, find the trace of the matrix. The trace of an n n matrix A is the sum of the main diagonal entries. That is, \(Tr(A) = a_{11} + a_{22} + \cdot + a_{nn}\). \(\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \) Text Transcription: Tr(A) = a_11 + a_22 + cdot + a_nn [_0^0^1 _0^1^0 _1^0^0]

In Exercises 63–66, find the trace of the matrix. The trace of an n n matrix A is the sum of the main diagonal entries. That is, \(Tr(A) = a_{11} + a_{22} + \cdot + a_{nn}\). \(\left[\begin{array}{rrrr} 1 & 0 & 2 & 1 \\ 0 & 1 & -1 & 2 \\ 4 & 2 & 1 & 0 \\ 0 & 0 & 5 & 1 \end{array}\right] \) Text Transcription: Tr(A) = a_11 + a_22 + cdot + a_nn [_0^4^0^1 _0^2^1^0 _5^1^-1^2 _1^0^2^1]

Trace of a Matrix In Exercises 6366, find the trace of the matrix. The trace of an n n matrix A is the sum of the main diagonal entries. That is, Tr(A) = a11 + a22 + . . . + ann. [ 1 4 3 2 4 0 6 1 3 6 2 1 2 1 1 3 ]

Prove that each statement is true when A and B are square matrices of order n and c is a scalar. (a) Tr(A + B) = Tr(A) + Tr(B) (b) Tr(cA) = cTr(A)

Prove that if A and B are square matrices of order n, then Tr(AB) = Tr(BA).

Find conditions on w, x, y, and z such that AB = BA for the matrices below. \(A=\left[\begin{array}{ll} w & x \\ y & z \end{array}\right] \) and \(B=\left[\begin{array}{rr} 1 & 1 \\ -1 & 1 \end{array}\right] \) Text Transcription: A = [_y^w _z^x] B = [_-1^1 _1^1]

Verify AB = BA for the matrices below. \(A=\left[\begin{array}{rr} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right] \) and \(B=\left[\begin{array}{rr} \cos \beta & -\sin \beta \\ \sin \beta & \cos \beta \end{array}\right] \) Text Transcription: A = [_sin a^ cos a _cos a ^-sin^a] B = [sin B ^ cos b cos B ^ -sin B]

Show that the matrix equation has no solution. \(\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right] A=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \) Text Transcription: [_1^1 _1^1] A = [_0^1 _1^0]

Show that no 2 × 2 matrices A and B exist that satisfy the matrix equation \(A B-B A=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \). Text Transcription: AB - Ba = [_0^1 _1^0]

Exploration Let i = 1 and let A = [ i 0 0 i ] and B = [ 0 i i 0]. (a) Find A2, A3, and A4. (Note: A2 = AA, A3 = AAA = A2A, and so on.) Identify any similarities with i 2, i 3, and i 4. (b) Find and identify B2.

Prove that if the product AB is a square matrix, then the product BA is defined. Getting Started: To prove that the product BA is defined, you need to show that the number of columns of B equals the number of rows of A. (i) Begin your proof by noting that the number of columns of A equals the number of rows of B. (ii) Then assume that A has size m × n and B has size n × p. (iii) Use the hypothesis that the product AB is a square matrix.

Prove that if both products AB and BA are defined, then AB and BA are square matrices.

Let A and B be matrices such that the product AB is defined. Show that if A has two identical rows, then the corresponding two rows of AB are also identical.

Let A and B be n × n matrices. Show that if the ith row of A has all zero entries, then the ith row of AB will have all zero entries. Give an example using 2 × 2 matrices to show that the converse is not true.

Let matrices A and B be of sizes 3 × 2 and 2 × 2, respectively. Answer each question and explain your answers. (a) Is it possible that A = B? (b) Is A + B defined? (c) Is AB defined? If so, is it possible that AB = BA?

A fruit grower raises two crops, apples and peaches. The grower ships each of these crops to three different outlets. In the matrix \(A=\left[\begin{array}{rrr} 125 & 100 & 75 \\ 100 & 175 & 125 \end{array}\right] \) \(a_{ij}\) represents the number of units of crop i that the grower ships to outlet j. The matrix B = [$3.50 $6.00] represents the profit per unit. Find the product BA and state what each entry of the matrix represents. Text Transcription: A = [_100^125 _175^100 _125^75] a_ij

A corporation has three factories, each of which manufactures acoustic guitars and electric guitars. In the matrix \(A=\left[\begin{array}{rrr} 70 & 50 & 25 \\ 35 & 100 & 70 \end{array}\right] \) \(a_{ij}\) represents the number of guitars of type i produced at factory j in one day. Find the production levels when production increases by 20%. Text Transcription: A = [_35^70 _100^50 _70^25] a_ij

In the matrix each entry \(p_{i j}(i \neq j)\) represents the proportion of the voting population that changes from party j to party i, and \(p_{ii}\) represents the proportion that remains loyal to party i from one election to the next. Find and interpret the product of P with itself. Text Transcription: p_ij(i neq j) p_ii

The matrices show the numbers of people (in thousands) who lived in each region of the United States in 2010 and 2013. The regional populations are separated into three age categories. (Source: U.S. Census Bureau) (a) The total population in 2010 was approximately 309 million and the total population in 2013 was about 316 million. Rewrite the matrices to give the information as percents of the total population. (b) Write a matrix that gives the changes in the percents of the population in each region and age group from 2010 to 2013. (c) Based on the result of part (b), which age group(s) show relative growth from 2010 to 2013?

In Exercises 83 and 84, perform the block multiplication of matrices A and B. If matrices A and B are each partitioned into four submatrices \(A=\left[\begin{array}{ll} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right] \) and \(B=\left[\begin{array}{ll} B_{11} & B_{12} \\ B_{21} & B_{22} \end{array}\right] \) then you can block multiply A and B, provided the sizes of the submatrices are such that the matrix multiplications and additions are defined. \(A B=\left[\begin{array}{ll} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right]\left[\begin{array}{ll} B_{11} & B_{12} \\ B_{21} & B_{22} \end{array}\right] \) \(=\left[\begin{array}{ll} A_{11} B_{11}+A_{12} B_{21} & A_{11} B_{12}+A_{12} B_{22} \\ A_{21} B_{11}+A_{22} B_{21} & A_{21} B_{12}+A_{22} B_{22} \end{array}\right] \) \(A=\left[\begin{array}{ll|ll} 1 & 2 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \hline 0 & 0 & 2 & 1 \end{array}\right], \quad B=\left[\begin{array}{rr|r} 1 & 2 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 3 \end{array}\right] \) Text Transcription: A = [_A_21 ^ A_11 _A_22 ^ A_12] B = [_B_21 ^ B_11 _B_22 ^ B_12] AB = [_A_21 ^ A-11 _A_22 ^B_11 _B_22 ^B_12] =[_A_21 B_11 + A_22 B_21 ^A_11 B_11 + A_12 B_21 _A_21 B_12 + A_22 B_22 ^ A_11 B_12 + A_12 B_22] A = [ _0^0^1 _0^1^2 _2^0^0 _1^0^0], B = [_0^0^-1^1 _0^0^1^2 _3^1^0^0]

Block Multiplication In Exercises 83 and 84, perform the block multiplication of matrices A and B. If matrices A and B are each partitioned into four submatrices A = [ A11 A21 A12 A22] and B = [ B11 B21 B12 B22] then you can block multiply A and B, provided the sizes of the submatrices are such that the matrix multiplications and additions are defined. AB = [ A11 A21 A12 A22] [ B11 B21 B12 B22] = [ A11B11 + A12B21 A21B11 + A22B21 A11B12 + A12B22 A21B12 + A22B22] A = [ 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 ] , B = [ 1 5 1 5 2 6 2 6 3 7 3 7 4 8 4 8 ]

In Exercises 85 and 86, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) For the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix. (b) The system Ax = b is consistent if and only if b can be expressed as a linear combination of the columns of A, where the coefficients of the linear combination are a solution of the system.

In Exercises 85 and 86, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) If A is an m × n matrix and B is an n × r matrix, then the product AB is an m × r matrix. (b) The matrix equation Ax = b, where A is the coefficient matrix and x and b are column matrices, can be used to represent a system of linear equations.

Evaluating an Expression In Exercises 16, evaluate the expression