Let \(A = [a_{ij}]_{n \times n}\) and \(B = [b_{ij}]_{n \times n}\). Then which of the following is/are true?
(A) \(AB = BA\)
(B) \((AB)^{-1} = B^{-1} A^{-1}\)
(C) \((AB)^T = B^T A^T\)
(D) \(AB = 0 \Rightarrow A = 0\) or \(B = 0\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (C) only
(A) \(AB = BA\) — Incorrect. Matrix multiplication is not commutative. For example,
\(\displaystyle A = \begin{bmatrix}0&1\\0&0\end{bmatrix},\ B = \begin{bmatrix}0&0\\1&0\end{bmatrix}\)
\(\displaystyle AB = \begin{bmatrix}1&0\\0&0\end{bmatrix},\ BA = \begin{bmatrix}0&0\\0&1\end{bmatrix}\)
(B) \((AB)^{-1} = B^{-1} A^{-1}\) — Correct. This is the reversal law for inverses. \((AB)^{-1}\) exists only when \(|AB| = |A||B| \ne 0\), that is, when A and B are both invertible, and then
\((AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = AIA^{-1} = I\)
So whenever the left side exists, it equals \(B^{-1}A^{-1}\).
(C) \((AB)^T = B^T A^T\) — Correct. This is the reversal law for transposes, true for all A and B of order n: the (i, j) entry of both sides is \(\sum_k a_{jk}b_{ki}\).
(D) \(AB = 0 \Rightarrow A = 0\) or \(B = 0\) — Incorrect. Two non-zero matrices can have a zero product:
\(\displaystyle A = \begin{bmatrix}1&0\\0&0\end{bmatrix},\ B = \begin{bmatrix}0&0\\0&1\end{bmatrix},\ AB = \begin{bmatrix}0&0\\0&0\end{bmatrix}\)