If A and B are invertible matrices, which of the following statement is not correct.
(a) $adj A = |A| A^{- 1}$
(b) $\det (A^{- 1}) = {(\det A)}^{- 1}$
(c) ${(A + B)}^{- 1} = A^{- 1} + B^{- 1}$
(d) ${(A B)}^{- 1} = B^{- 1} A^{- 1}$

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Solution

(c) ${(A + B)}^{- 1} = A^{- 1} + B^{- 1}$

We have, $adj A = |A| A^{- 1}$ , $\det (A^{- 1}) = {(\det A)}^{- 1}$ and ${(A B)}^{- 1} = B^{- 1} A^{- 1}$ all are the properites of inverse of a matrix.

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Similar questions

|A|

If A and B are invertible matrices, then which of the following is not correct?

(a) $a d j A = | A | . A^{- 1}$
(b) $d e t (A)^{- 1} = [d e t (A)]^{- 1}$
(c) $(A B)^{- 1} = B^{- 1} A^{- 1}$
(d) $(A + B)^{- 1} = B^{- 1} + A^{- 1}$

[A (A + B)^{- 1} B]^{- 1} = B^{- 1} + A^{- 1}

[A (A + B)^{- 1} B]^{- 1} = [A (A^{- 1} + B^{- 1}) B]^{- 1} = [(I + A B^{- 1}) B]^{- 1} = [(B + A B^{- 1} B)]^{- 1} = [(B + A I)]^{- 1} = [(B + 1)]^{- 1} = B^{- 1} + A^{- 1}

A

B

1. d e t (A + B) = d e t A + d e t B

2. (A + B)^{- 1} = A^{- 1} + B^{- 1}

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