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.- 1False; if A and B are invertible matrices, then (AB)=B-A¯1.- 1O 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-1.1b. 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 Ais also invertible.a bc. If A =and ad = bc, then A is not invertible.c da bis invertible if and only if a +b and b+d.c dO A. True; A =B. False; if A is invertible, then ad = bc.1d -bOc. True; if ad = bc then ad - bc = 0, andis undefined.ad - bc- ca 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 tothe product E,'E,'E3O 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- 1the product E,•• •E, ¯'E, ¯'E, -1.OC.- 1True; 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 Amay be reduced to the identity.

As per guidelines, we will solve only first three subparts. Kindly repost other subparts separately.

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ