Consider the following in respect of non-singular matrices $A$ and $B$ :
I. $(AB)^{-1} = A^{-1}B^{-1}$
II. $(BA)(AB)^{-1} = I$ , where $I$ is the identity matrix
III. $(AB)^T = A^T B^T$
How many of the above are correct?

Question

Consider the following in respect of non-singular matrices

A

and

B

:
I.

(AB)^{-1} = A^{-1}B^{-1}

II.

(BA)(AB)^{-1} = I

, where

I

is the identity matrix
III.

(AB)^T = A^T B^T

How many of the above are correct?

Quantrex Academy · Accepted Answer

Correct answer: A. None

For Statement I, the reversal law of matrix inverse gives

(AB)^{-1} = B^{-1}A^{-1}

. Since matrix multiplication is not commutative in general,

B^{-1}A^{-1} \neq A^{-1}B^{-1}

. Thus, Statement I is incorrect.

For Statement II, using the reversal law,

(BA)(AB)^{-1} = (BA)(B^{-1}A^{-1}) = B(AB^{-1})A^{-1} = BIA^{-1} = BA^{-1}

. This is not equal to

I

in general. Thus, Statement II is incorrect.

For Statement III, the reversal law of matrix transpose gives

(AB)^T = B^T A^T

. Since matrix multiplication is not commutative,

B^T A^T \neq A^T B^T

in general. Thus, Statement III is incorrect.

Since none of the statements are correct, the number of correct statements is zero.

Answer: None

Consider the following in respect of non-singular matrices A and B...