Let A, B and C be square matrices of the same order, and let O be the zero matrix of that order. Consider the following statements:
(A) If \(AB = O\), then \(A = O\) or \(B = O\).
(B) \((A + B)^2 = A^2 + 2AB + B^2\) for all such matrices A and B.
(C) \(A(B + C) = AB + AC\)
(D) If \(AB = BA\), then \((A + B)(A - B) = A^2 - B^2\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (C) and (D) only
Matrix multiplication is distributive over addition, but it is not commutative, so AB and BA cannot be combined unless they are equal. Expanding:
\((A + B)^2 = A^2 + AB + BA + B^2\)
\((A + B)(A - B) = A^2 - AB + BA - B^2\)
For (A), take
\( \displaystyle A = \begin{bmatrix}1&0\\0&0\end{bmatrix},\quad B = \begin{bmatrix}0&0\\0&1\end{bmatrix},\quad AB = \begin{bmatrix}0&0\\0&0\end{bmatrix} \)
So the product of two non-zero matrices can be O, \((A + B)^2\) has the middle part \(AB + BA\), and when AB = BA the terms \(-AB + BA\) cancel.
Checking the statements
(A) If \(AB = O\), then \(A = O\) or \(B = O\). — Incorrect. In the example above AB = O but neither A nor B is O.
(B) \((A + B)^2 = A^2 + 2AB + B^2\) for all such matrices A and B. — Incorrect. The middle part is \(AB + BA\), which equals 2AB only when AB = BA.
(C) \(A(B + C) = AB + AC\) — Correct. This is the distributive law of matrix multiplication.
(D) If \(AB = BA\), then \((A + B)(A - B) = A^2 - B^2\). — Correct. With AB = BA, \(-AB + BA = O\).