Question

please tell me this answer​

Answer 1

$\green{\large\underline{\sf{Solution-}}}$

Given matrices are

$\rm :\longmapsto\:A = \begin{bmatrix} 2 & 3\\ 4 & - 1\end{bmatrix}$

and

$\rm :\longmapsto\:B = \begin{bmatrix} 1 & - 2\\ - 1 & 3\end{bmatrix}$

So, Consider

$\red{\rm :\longmapsto\: {A}^{2}}$

$\rm \:  =  \: A \times A$

$\rm \:  =  \: \begin{bmatrix} 2 & 3\\ 4 & - 1\end{bmatrix} \times \begin{bmatrix} 2 & 3\\ 4 & - 1\end{bmatrix}$

$\rm \:  =  \: \begin{bmatrix} 4 + 12 & 6 - 3\\ 8 - 4 & 12 + 1\end{bmatrix}$

$\rm \:  =  \: \begin{bmatrix} 16 & 3\\ 4 & 13\end{bmatrix}$

$\rm \implies\: {A}^{2} =  \: \begin{bmatrix} 16 & 3\\ 4 & 13\end{bmatrix}$

Now, Consider

$\red{\rm :\longmapsto\: {B}^{2}}$

$\rm \:  =  \: B \times B$

$\rm \:  =  \: \begin{bmatrix} 1 & - 2\\ - 1 & 3\end{bmatrix} \times \begin{bmatrix} 1 & - 2\\ - 1 & 3\end{bmatrix}$

$\rm \:  =  \: \begin{bmatrix} 1 + 2 & - 2 - 6\\ - 1 - 3 & 2 + 9\end{bmatrix}$

$\rm \:  =  \: \begin{bmatrix} 3 & - 8\\ - 4 & 11\end{bmatrix}$

$\bf\implies \: {B}^{2}   =  \: \begin{bmatrix} 3 & - 8\\ - 4 & 11\end{bmatrix}$

Now, Consider

$\red{\rm :\longmapsto\: {A}^{2} - {B}^{2}}$

$\rm \:  =  \: \begin{bmatrix} 16 & 3\\ 4 & 13\end{bmatrix} - \begin{bmatrix} 3 & - 8\\ - 4 & 11\end{bmatrix}$

$\rm \:  =  \: \begin{bmatrix} 13 & 11\\ 8 & 2\end{bmatrix}$

$\rm \implies\: \boxed{\tt{ {A}^{2} - {B}^{2} =  \: \begin{bmatrix} 13 & 11\\ 8 & 2\end{bmatrix}}}$

• Hence, option (2) is correct.

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More to know :-

Multiplication of matrices is defined when number of columns of pre multiplier is equal to the number of rows of post multiplier.

Matrix multiplication may or may not be Commutative

Matrix multiplication is Associative

Matrix multiplication is Distributive