C In Exercises 8386, find A1 if A is invertible.A =5 2 1 02 3142 1 3 23 5 2 4

In Exercises 14, use the Quick Formula to find the inverse of the given matrix, if it exists.  7 3 2 1

In Exercises 14, use the Quick Formula to find the inverse of the given matrix, if it exists.  5 2 4 3

In Exercises 14, use the Quick Formula to find the inverse of the given matrix, if it exists.  2 5 4 10

In Exercises 14, use the Quick Formula to find the inverse of the given matrix, if it exists.  6 2 5 1

In Exercises 516, use an augmented matrix and row operations to find the inverse of the given matrix, if it exists.  1 4 2 9

In Exercises 516, use an augmented matrix and row operations to find the inverse of the given matrix, if it exists.  4 13 1 3

In Exercises 516, use an augmented matrix and row operations to find the inverse of the given matrix, if it exists. 101 010 111

In Exercises 516, use an augmented matrix and row operations to find the inverse of the given matrix, if it exists. 011 010 101

In Exercises 516, use an augmented matrix and row operations to find the inverse of the given matrix, if it exists. 1 2 1 01 3 00 1

In Exercises 516, use an augmented matrix and row operations to find the inverse of the given matrix, if it exists.  1 2 1 4 7 7 1 1 5

In Exercises 516, use an augmented matrix and row operations to find the inverse of the given matrix, if it exists.  1 3 1 2 5 4 2 3 8 

In Exercises 516, use an augmented matrix and row operations to find the inverse of the given matrix, if it exists. 3 1 9 1 1 4 2 2 10

In Exercises 516, use an augmented matrix and row operations to find the inverse of the given matrix, if it exists. 0010 1000 0001 0100

In Exercises 516, use an augmented matrix and row operations to find the inverse of the given matrix, if it exists. 100 2 0 1 1 0 0 230 200 3

In Exercises 516, use an augmented matrix and row operations to find the inverse of the given matrix, if it exists. 13 1 4 0 1 2 2 00 1 1 00 0 1

In Exercises 516, use an augmented matrix and row operations to find the inverse of the given matrix, if it exists. 1 3 1 2 2 5 4 2 3 9 2 5 4 12 4 7

Use the answer to Exercise 6 to find the solutions to the linear system 4x1 + 13x2 = 3 x1 + 3x2 = 2

Use the answer to Exercise 10 to find the solutions to the linear system x1 + 2x2 x3 = 2 4x1 7x2 + 7x3 = 1 x1 x2 + 5x3 = 1

Use the answer to Exercise 12 to find the solutions to the linear system 3x1 x2 + 9x3 = 4 x1 x2 + 4x3 = 1 2x1 2x2 + 10x3 = 3

Use the answer to Exercise 14 to find the solutions to the linear system x1 2x4 = 1 x2 x3 = 2 2x2 + 3x3 = 2 2x1 3x4 = 1

In Exercises 2126, determine if the given linear transformation T is invertible, and if so, find T1. (HINT: Start by finding the matrix A such that T(x) = Ax.) T x1 x2  =  4x1 + 3x2 3x1 + 2x2 

In Exercises 2126, determine if the given linear transformation T is invertible, and if so, find T1. (HINT: Start by finding the matrix A such that T(x) = Ax.) T x1 x2  = 2x1 5x2 x1 + 4x2 x1 + x2 

In Exercises 2126, determine if the given linear transformation T is invertible, and if so, find T1. (HINT: Start by finding the matrix A such that T(x) = Ax.) T x1 x2  =  x1 5x2 2x1 + 10x2

In Exercises 2126, determine if the given linear transformation T is invertible, and if so, find T1. (HINT: Start by finding the matrix A such that T(x) = Ax.) T x1 x2 x3  =  x1 + x3 x2 x3 x1 x2 + x3 

In Exercises 2126, determine if the given linear transformation T is invertible, and if so, find T1. (HINT: Start by finding the matrix A such that T(x) = Ax.) T x1 x2 x3  =  x1 + 2x2 x3 x1 + x2 x3 

In Exercises 2126, determine if the given linear transformation T is invertible, and if so, find T1. (HINT: Start by finding the matrix A such that T(x) = Ax.) T x1 x2 x3  = x1 + x2 x3 x2 x3 x1 x2 + x3

Let T1 and T2 be linear transformations given by T1 x1 x2  =  2x1 + x2 x1 + x2  T2 x1 x2  =  3x1 + 2x2 x1 + x2  Find the matrix A such that (a) T1 1 (T2(x)) = Ax (b) T1(T1 2 (x)) = Ax (c) T1 2 (T1(x)) = Ax (d) T2(T1 1 (x)) = Ax

Let T1 and T2 be linear transformations given by T1 x1 x2  =  3x1 + 5x2 4x1 + 7x2  T2 x1 x2  =  2x1 + 9x2 x1 + 5x2  Find the matrix A such that (a) T1 1 (T2(x)) = Ax (b) T1(T1 2 (x)) = Ax (c) T1 2 (T1(x)) = Ax (d) T2(T1 1 (x)) = Ax

In Exercises 2934, use an appropriate partitioning of the matrix A to find A1. A = 100 027 014

In Exercises 2934, use an appropriate partitioning of the matrix A to find A1. A = 520 210 001

In Exercises 2934, use an appropriate partitioning of the matrix A to find A1. A = 2500 3800 0014 0013

In Exercises 2934, use an appropriate partitioning of the matrix A to find A1. A = 1300 3800 1225 4313

In Exercises 2934, use an appropriate partitioning of the matrix A to find A1. A = 1 300 0 3 800 0 1 212 2 4 301 0 1 200 1

In Exercises 2934, use an appropriate partitioning of the matrix A to find A1. A = 7 200 0 4 100 0 1 310 0 2 301 2 5 203 5

FIND AN EXAMPLE For Exercises 3540, find an example that meets the given specifications. A diagonal 3 3 invertible matrix A.

FIND AN EXAMPLE For Exercises 3540, find an example that meets the given specifications. A singular 3 3 matrix A that has no zero entries.

FIND AN EXAMPLE For Exercises 3540, find an example that meets the given specifications. 2 2 matrices A and B such that AB = 3I2.

FIND AN EXAMPLE For Exercises 3540, find an example that meets the given specifications. 3 3 matrices A and B such that B A = 2I2.

FIND AN EXAMPLE For Exercises 3540, find an example that meets the given specifications. A 23 matrix A and a 32 matrix B such that AB = I2 but B A = I3.

FIND AN EXAMPLE For Exercises 3540, find an example that meets the given specifications. A 34 matrix A and a 43 matrix B such that AB = I3 but B A = I4.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. If A is an invertible nn matrix, then the number of solutions to Ax = b depends on the vector b in Rn.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. A must be a square matrix to be invertible.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. If an n n matrix A is equivalent to In, then A1 is also equivalent to In.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. If an n n matrix A is singular, then the columns of A must be linearly independent.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. The Caesar cipher encoding system is an example of a linear transformation.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. If the columns of an nn matrix AspanRn, then Ais singular.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. If A and B are invertible n n matrices, then the inverse of AB is B1A1.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. If A and B are invertible n n matrices, then the inverse of A + B is A1 + B1.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. If A is invertible, then  A1 1 = A

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. If AB = 2I3, then B A = 2I3.

In Exercises 5154, solve for the matrix X. Assume that all matrices are n n and invertible as needed. AX = B

In Exercises 5154, solve for the matrix X. Assume that all matrices are n n and invertible as needed. B X = A + C X

In Exercises 5154, solve for the matrix X. Assume that all matrices are n n and invertible as needed. B(X + A)1 = C

In Exercises 5154, solve for the matrix X. Assume that all matrices are n n and invertible as needed. AX(D + B X)1 = C

Find all 2 2 matrices A such that A1 = A

Suppose that A is a square matrix with two equal rows. Is A invertible? Justify your answer.

Suppose that A is a square matrix with two equal columns. Is A invertible? Justify your answer.

Let A =  a 0 0 d  be a 2 2 diagonal matrix. For what values of a and d will A be invertible?

For what values of c will A =  1 1 c c 2  be invertible?

. Suppose that A1 =  4 6 2 14 . Find (2A)1.

Let A be an n n matrix and b be in Rn. If Ax = b has a unique solution, show that A must be invertible.

Suppose that A = PDP 1, where all matrices are square. Find an expression for each of A2 and A3, and then give a general formula for An.

Suppose that A, B, and C are n n invertible matrices. Solve AC = C B for B.

Suppose that A is an invertible n n matrix and that X and B are n m matrices. If AX = B, prove that X = A1B.

Suppose that A is an invertible m m matrix and that B and C are n m matrices. If (B C)A = 0nm, prove that B = C.

Let A and B be n n matrices, and suppose that B and AB are both invertible. Prove that A is also invertible.

Let A and B be nn matrices. Prove that if B is singular, then so is AB.

Let A and B be n n matrices. Prove that if A is singular, then so is AB.

Complete the proof of Theorem 3.19: If T is an invertible linear transformation, then T1(r x) = r T1(x).

Complete the proof of Theorem 3.23: Let A be an invertible n n matrix and C and D be n m matrices. (a) If AC = AD, then C = D. (b) If AC = 0nm, then C = 0nm.

C For Exercises 7174, refer to the MP3 scenario given at the beginning of the Section 3.1. There a linear transformation T(x) is defined that gives the costs associated with a production vector x. Determine T1, and use it to find the production level for each type of player that will result in the given costs. If the given costs are not possible, explain why. . Labor = $1070, Materials = $1002, and Overhead = $1368.

C For Exercises 7174, refer to the MP3 scenario given at the beginning of the Section 3.1. There a linear transformation T(x) is defined that gives the costs associated with a production vector x. Determine T1, and use it to find the production level for each type of player that will result in the given costs. If the given costs are not possible, explain why. Labor = $1148, Materials = $1076, and Overhead = $1452.

C For Exercises 7174, refer to the MP3 scenario given at the beginning of the Section 3.1. There a linear transformation T(x) is defined that gives the costs associated with a production vector x. Determine T1, and use it to find the production level for each type of player that will result in the given costs. If the given costs are not possible, explain why. Labor = $2045, Materials = $1965, and Overhead = $2615.

C For Exercises 7174, refer to the MP3 scenario given at the beginning of the Section 3.1. There a linear transformation T(x) is defined that gives the costs associated with a production vector x. Determine T1, and use it to find the production level for each type of player that will result in the given costs. If the given costs are not possible, explain why. Labor = $2348, Materials = $2200, and Overhead = $2983.

C Suppose that you are in the garden supply business. Naturally, one of the things that you sell is fertilizer. You have three brands available: Vigoro and Parkers as introduced in Section 2.1, and a third brand, Bleyers SuperRich. The amount of nitrogen, phosphoric acid, and potash per 100 pounds for each brand is given by the nutrient vectors v = 29 3 4  p = 18 25 6  b = 50 19 9  Vigoro Parkers Bleyers In Exercises 7578, determine the linear transformation T : R3 R3 that takes a vector of brand amounts (in hundreds of pounds) as input and gives the nutrient vector as output. Then find a formula for T1, and use it to determine the amount of Vigoro, Parkers, and Bleyers required to produce the specified nutrient mix. 409 pounds of nitrogen, 204 pounds of phosphoric acid, and 81 pounds of potash.

C Suppose that you are in the garden supply business. Naturally, one of the things that you sell is fertilizer. You have three brands available: Vigoro and Parkers as introduced in Section 2.1, and a third brand, Bleyers SuperRich. The amount of nitrogen, phosphoric acid, and potash per 100 pounds for each brand is given by the nutrient vectors v = 29 3 4  p = 18 25 6  b = 50 19 9  Vigoro Parkers Bleyers In Exercises 7578, determine the linear transformation T : R3 R3 that takes a vector of brand amounts (in hundreds of pounds) as input and gives the nutrient vector as output. Then find a formula for T1, and use it to determine the amount of Vigoro, Parkers, and Bleyers required to produce the specified nutrient mix. 439 pounds of nitrogen, 147 pounds of phosphoric acid, and 76 pounds of potash.

C Suppose that you are in the garden supply business. Naturally, one of the things that you sell is fertilizer. You have three brands available: Vigoro and Parkers as introduced in Section 2.1, and a third brand, Bleyers SuperRich. The amount of nitrogen, phosphoric acid, and potash per 100 pounds for each brand is given by the nutrient vectors v = 29 3 4  p = 18 25 6  b = 50 19 9  Vigoro Parkers Bleyers In Exercises 7578, determine the linear transformation T : R3 R3 that takes a vector of brand amounts (in hundreds of pounds) as input and gives the nutrient vector as output. Then find a formula for T1, and use it to determine the amount of Vigoro, Parkers, and Bleyers required to produce the specified nutrient mix. 1092 pounds of nitrogen, 589 pounds of phosphoric acid, and 223 pounds of potash.

C Suppose that you are in the garden supply business. Naturally, one of the things that you sell is fertilizer. You have three brands available: Vigoro and Parkers as introduced in Section 2.1, and a third brand, Bleyers SuperRich. The amount of nitrogen, phosphoric acid, and potash per 100 pounds for each brand is given by the nutrient vectors v = 29 3 4  p = 18 25 6  b = 50 19 9  Vigoro Parkers Bleyers In Exercises 7578, determine the linear transformation T : R3 R3 that takes a vector of brand amounts (in hundreds of pounds) as input and gives the nutrient vector as output. Then find a formula for T1, and use it to determine the amount of Vigoro, Parkers, and Bleyers required to produce the specified nutrient mix. 744 pounds of nitrogen, 428 pounds of phosphoric acid, and 156 pounds of potash.

C In Exercises 7982, the given set of vectors are the encoded version of a short message. Decode the message, given that the encoding matrix is A =  1 3 2 2 7 9 4 14 17 $ 41 161 306 ,  7 79 142%

C In Exercises 7982, the given set of vectors are the encoded version of a short message. Decode the message, given that the encoding matrix is A =  1 3 2 2 7 9 4 14 17 $30 45 96 ,  1 22 39%

C In Exercises 7982, the given set of vectors are the encoded version of a short message. Decode the message, given that the encoding matrix is A =  1 3 2 2 7 9 4 14 17 $ 7 75 136 , 25 37 79 ,  47 158 303%

C In Exercises 7982, the given set of vectors are the encoded version of a short message. Decode the message, given that the encoding matrix is A =  1 3 2 2 7 9 4 14 17 $ 44 157 300 ,  46 105 211 ,  23 88 167 ,  40 152 289%

C In Exercises 8386, find A1 if A is invertible. A = 3 1 2 0 22 51 3 0 2 2 41 23

C In Exercises 8386, find A1 if A is invertible. A = 5 2 1 0 2 314 2 1 3 2 3 5 2 4

C In Exercises 8386, find A1 if A is invertible. A = 5 1 2 12 3 2 2 10 2 3 1 01 5 1 1 1 3 0 0 3 21

C In Exercises 8386, find A1 if A is invertible. A = 22904 95521 23005 95070 49951