If A and B are two square matrices of the same order such that (A + B)2 = A2 + B2 ,then
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → AB = -BA
Matrix multiplication is not commutative, so AB and BA must be kept separate when expanding:
\( \displaystyle (A+B)^2=(A+B)(A+B) \)
\( \displaystyle =A^2+AB+BA+B^2 \)
Putting this equal to \( A^2+B^2 \):
\( \displaystyle AB+BA=O \)
\( \displaystyle AB=-BA \)
Why the other options are incorrect
Option (1) A2 + B2 = O — Incorrect. Nothing forces \( A^2+B^2=O \); for example A = I and B = O satisfy the condition, but \( A^2+B^2=I \).
Option (2) AB = O — Incorrect. The condition gives only AB + BA = O. For \( A=\begin{bmatrix}1&0\\0&-1\end{bmatrix} \) and \( B=\begin{bmatrix}0&1\\1&0\end{bmatrix} \), AB + BA = O, but \( AB=\begin{bmatrix}0&1\\-1&0\end{bmatrix}\ne O \).
Option (3) BA = O — Incorrect. For the same pair, \( BA=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\ne O \).