If A and B are two non-singular matrices of same order, then which of…

The correct answer is Option (1) → B and D only

A and B are non-singular, so |A| ≠ 0 and |B| ≠ 0. Then

    \( \displaystyle |\mathrm{AB}|=|\mathrm{A}|\,|\mathrm{B}|\ne 0 \)

so AB is invertible, with \( (\mathrm{AB})^{-1}=\mathrm{B}^{-1}\mathrm{A}^{-1} \). The adjoint of a product, like the inverse, reverses the order:

    \( \displaystyle \mathrm{adj}(\mathrm{AB})=\mathrm{adj}\,\mathrm{B}\cdot\mathrm{adj}\,\mathrm{A} \)

The inverse of a sum is not the sum of the inverses. For example, with A = B = I:

    \( \displaystyle (\mathrm{A}+\mathrm{B})^{-1}=(2\mathrm{I})^{-1}=\frac{1}{2}\mathrm{I} \)

    \( \displaystyle \mathrm{B}^{-1}+\mathrm{A}^{-1}=2\mathrm{I} \)

So |AB| ≠ 0 (AB is non-singular and invertible), adj (AB) = adj B . adj A, and \( (\mathrm{A}+\mathrm{B})^{-1} \) is in general not \( \mathrm{B}^{-1}+\mathrm{A}^{-1} \).

Checking the statements

(A) AB is non singular. — Correct. \( |\mathrm{AB}|=|\mathrm{A}|\,|\mathrm{B}|\ne 0 \).

(B) \( (\mathrm{A}+\mathrm{B})^{-1}=\mathrm{B}^{-1}+\mathrm{A}^{-1} \). — Incorrect. With A = B = I the left side is \( \frac{1}{2}\mathrm{I} \) and the right side is 2I.

(C) adj (AB) = adj B. adj A. — Correct. The adjoint of a product reverses the order.

(D) AB is not invertible. — Incorrect. |AB| ≠ 0, so AB is invertible.

The statements that are not correct are (B) and (D).