Q. 344.5( 2 Votes )

If A and B are in A. adj A = |A|. A-1

B. det (A)-1 = [det (A)]-1

C. (AB)-1 = B-1 A-1

D. (A + B)-1 = B-1 + A-1

Answer :

Given A and B are invertible matrices.


Consider (AB) B-1 A-1


(AB) B-1 A-1 = A(BB-1) A-1


= AIA-1 = (AI) A-1


= AA-1 = I


(AB)-1 = B-1 A-1 … option (C)


Also AA-1 = I


|AA-1| = |I|


|A| |A-1| = 1



det (A)-1 = [det (A)]-1 … (B)


We know that


adj A = |A|. A-1 … option (A)



But


(A + B)-1 ≠ B-1 + A-1


Hence, option (D) is not correct.

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