Mark each statement True or False. Justify each answer. Complete parts (a) through (e) below. ..... a. A product of invertible nxn matrices is invertible, and the inverse of the product is the product of their inverses in the same order. O A. True; since invertible matrices commute, (AB) 1= B-A-1 =A-B-1. В. False; if A and B are invertible matrices, then (AB) 1 = B-A-1. - 1 O C. False; if A and B are invertible matrices, then (AB = BA-B1. O D. True; if A and B are invertible matrices, then (AB)-1 = =A'B-1. 1 b. If A is invertible, then the inverse of A is A itself. O A. False; since inverses are not unique, it is possible that B#A is the inverse of A1. O B. True; since A-1 is the inverse of A, A-1A =1=AA-1. since A-A =1= AA 1, A is the inverse of A-1. O C. True; A is invertible if and only if A = A1. Since A is invertible, A = A1. Since A = A1, A-1 is invertible, and A is the inverse of A 1. O D. False; it does not follow from the fact that A is invertible that A is also invertible. a b c. If A = and ad = bc, then A is not invertible. c d a b is invertible if and only if a +b and b+d. c d O A. True; A = B. False; if A is invertible, then ad = bc. 1 d -b OC. True; if ad = bc then ad - bc = 0, and is undefined. ad - bc - c a
Mark each statement True or False. Justify each answer. Complete parts (a) through (e) below. ..... a. A product of invertible nxn matrices is invertible, and the inverse of the product is the product of their inverses in the same order. O A. True; since invertible matrices commute, (AB) 1= B-A-1 =A-B-1. В. False; if A and B are invertible matrices, then (AB) 1 = B-A-1. - 1 O C. False; if A and B are invertible matrices, then (AB = BA-B1. O D. True; if A and B are invertible matrices, then (AB)-1 = =A'B-1. 1 b. If A is invertible, then the inverse of A is A itself. O A. False; since inverses are not unique, it is possible that B#A is the inverse of A1. O B. True; since A-1 is the inverse of A, A-1A =1=AA-1. since A-A =1= AA 1, A is the inverse of A-1. O C. True; A is invertible if and only if A = A1. Since A is invertible, A = A1. Since A = A1, A-1 is invertible, and A is the inverse of A 1. O D. False; it does not follow from the fact that A is invertible that A is also invertible. a b c. If A = and ad = bc, then A is not invertible. c d a b is invertible if and only if a +b and b+d. c d O A. True; A = B. False; if A is invertible, then ad = bc. 1 d -b OC. True; if ad = bc then ad - bc = 0, and is undefined. ad - bc - c a
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Mark each statement True or False. Justify each answer. Complete parts (a) through (e) below.
.....
a. A product of invertible nxn matrices is invertible, and the inverse of the product is the product of their inverses in the same order.
O A. True; since invertible matrices commute, (AB)-1 = B-A-1=A-'B-1.
В.
- 1
False; if A and B are invertible matrices, then (AB)=B-A¯1.
- 1
O C. False; if A and B are invertible matrices, then (AB = BA-B1.
O D. True; if A and B are invertible matrices, then (AB)-1 =
=A'B-1.
1
b. If A is invertible, then the inverse of A is A itself.
O A. False; since inverses are not unique, it is possible that B#A is the inverse of A1.
O B. True; since A-1 is the inverse of A, A-1A =1=AA-1. since A-A =1= AA 1, A is the inverse of A-1.
O C. True; A is invertible if and only if A = A1. Since A is invertible, A = A1. Since A = A1, A-1 is invertible, and A is the inverse of A 1.
O D. False; it does not follow from the fact that A is invertible that A
is also invertible.
a b
c. If A =
and ad = bc, then A is not invertible.
c d
a b
is invertible if and only if a +b and b+d.
c d
O A. True; A =
B. False; if A is invertible, then ad = bc.
1
d -b
Oc. True; if ad = bc then ad - bc = 0, and
is undefined.
ad - bc
- c
a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8548185-9e31-4f1b-b220-ed2aa4852b02%2F043991a4-26d5-41c1-b177-1fb6218489eb%2F8vpihh_processed.png&w=3840&q=75)
Transcribed Image Text:Mark each statement True or False. Justify each answer. Complete parts (a) through (e) below.
.....
a. A product of invertible nxn matrices is invertible, and the inverse of the product is the product of their inverses in the same order.
O A. True; since invertible matrices commute, (AB)-1 = B-A-1=A-'B-1.
В.
- 1
False; if A and B are invertible matrices, then (AB)=B-A¯1.
- 1
O C. False; if A and B are invertible matrices, then (AB = BA-B1.
O D. True; if A and B are invertible matrices, then (AB)-1 =
=A'B-1.
1
b. If A is invertible, then the inverse of A is A itself.
O A. False; since inverses are not unique, it is possible that B#A is the inverse of A1.
O B. True; since A-1 is the inverse of A, A-1A =1=AA-1. since A-A =1= AA 1, A is the inverse of A-1.
O C. True; A is invertible if and only if A = A1. Since A is invertible, A = A1. Since A = A1, A-1 is invertible, and A is the inverse of A 1.
O D. False; it does not follow from the fact that A is invertible that A
is also invertible.
a b
c. If A =
and ad = bc, then A is not invertible.
c d
a b
is invertible if and only if a +b and b+d.
c d
O A. True; A =
B. False; if A is invertible, then ad = bc.
1
d -b
Oc. True; if ad = bc then ad - bc = 0, and
is undefined.
ad - bc
- c
a
![O D. False; if ad = bc, then A is invertible.
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 n.
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 to the identity would correspond to
the product E,'E,'E3
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
- 1
the product E,
•• •E, ¯'E, ¯'E, -1.
OC.
- 1
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
may be reduced to the identity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8548185-9e31-4f1b-b220-ed2aa4852b02%2F043991a4-26d5-41c1-b177-1fb6218489eb%2F88y0af5_processed.png&w=3840&q=75)
Transcribed Image Text:O D. False; if ad = bc, then A is invertible.
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 n.
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 to the identity would correspond to
the product E,'E,'E3
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
- 1
the product E,
•• •E, ¯'E, ¯'E, -1.
OC.
- 1
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
may be reduced to the identity.
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