d. If A can be row reduced to the identity matrix, then A must be invertible.O A. False; not every matrix that is row equivalent to the identity matrix is invertible.O B. False; since the identity matrix is not invertible, A is not invertible either.O C. True; since A can be row reduced to the identity matrix, A is row equivalent to the identity matrix. Since every matrix that is row equivalent to the identity is invertible, A is invertible.O D. True; since A can be row reduced to the identity matrix, I is the inverse of A. Therefore, A is invertible.e. If A is invertible, then elementary row operations that reduce A to the identity I, also reduce A-1 to I,.O A. False; if A is invertible, then the row operations required to reduce A to the identity correspond to some product of elementary matrices E, E,E3•• •En. Then the row operations required to reduce A- 1to the identity would correspond to-1the product E, 'E2 E,1...E,O B. False; if A is invertible, then the row operations required to reduce A to the identity correspond to some product of elementary matrices E, E,E3•• •En. Then the row operations required to reduce Ato the identity would correspond tothe product E,-1... E3- 1-1O C. True; if A is invertible then A is the product of some number of elementary matrices E, E,E3•.•En, each corresponding to row operations. Then Ais E,... E3E,E1, the same elementary matrices.O D. True: by using the same row operations in reversed order A- 1may be reduced to the identity. Mark each statement True or False. Justify each answer.

Since you have asked multiple questions so as per guidelines we will solve the first question for…

1. If A can be row reduced to the identity matrix, then A must be invertible. OA. False; not every matrix that is row equivalent to the identity matrix is invertible. OB. False; since the identity matrix is not invertible, A is not invertible either. O C. True; since A can be row reduced to the identity matrix, A is row equivalent to the identity matrix. Since every matrix that is row equivalent to the identity is invertible, A is invertible. O D. True; since A can be row reduced to the identity matrix, I is the inverse of A. Therefore, A is invertible.

1. If A can be row reduced to the identity matrix, then A must be invertible. OA. False; not every matrix that is row equivalent to the identity matrix is invertible. OB. False; since the identity matrix is not invertible, A is not invertible either. O C. True; since A can be row reduced to the identity matrix, A is row equivalent to the identity matrix. Since every matrix that is row equivalent to the identity is invertible, A is invertible. O D. True; since A can be row reduced to the identity matrix, I is the inverse of A. Therefore, A is invertible.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

d. If A can be row reduced to the identity matrix, then A must be invertible.
O A. False; not every matrix that is row equivalent to the identity matrix is invertible.
O B. False; since the identity matrix is not invertible, A is not invertible either.
O C. True; since A can be row reduced to the identity matrix, A is row equivalent to the identity matrix. Since every matrix that is row equivalent to the identity is invertible, A is invertible.
O D. True; since A can be row reduced to the identity matrix, I is the inverse of A. Therefore, A is invertible.
e. If A is invertible, then elementary row operations that reduce A to the identity I, also reduce A-1 to I,.
O A. False; if A is invertible, then the row operations required to reduce A to the identity correspond to some product of elementary matrices E, E,E3•• •En. Then the row operations required to reduce A
- 1
to the identity would correspond to
-1
the product E, 'E2 E,1...E,
O B. False; if A is invertible, then the row operations required to reduce A to the identity correspond to some product of elementary matrices E, E,E3•• •En. Then the row operations required to reduce A
to the identity would correspond to
the product E,
-1... E3
- 1
-1
O C. True; if A is invertible then A is the product of some number of elementary matrices E, E,E3•.•En, each corresponding to row operations. Then A
is E,... E3E,E1, the same elementary matrices.
O D. True: by using the same row operations in reversed order A
- 1
may be reduced to the identity.

Mark each statement True or False. Justify each answer.

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