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  4. If A=[0&1 0&0], I is the unit matrix of order 2 and a, b are arbitrary constants, then (aI + bA)2 is equal to

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Q. 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

Matrices

Solution:

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