Question

If $A=\begin{bmatrix}0&1\\\ 0&0\end{bmatrix}$, I is the unit matrix of order 2 and a, b are arbitrary constants, then $(aI + bA)^2$ is equal to

• A. $a^2I + abA$
• B. $a^2I + 2abA$
• C. $a^2I + b^2A$
• D. None of these

Answer 1

Answer: (B)

$a^2I + 2abA$

Answer 2

$\left(aI+bA\right)^{2}=a^{2}I^{2}+b^{2}A^{2}+2ab\,AI$ $=a^{2}I^{2}+b^{2}\,A^{2}+2abA$ But $A^{2}=\begin{bmatrix}0&0\\\ 0&0\end{bmatrix} \therefore \left(aI+bA\right)^{2}=a^{2}I+2abA.$