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MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES5 Marks
Question
If $\text{A}=\begin{bmatrix}0&-\text{x}\\\text{x}&0\end{bmatrix},\ \text{B}=\begin{bmatrix}0&1\\1&0\end{bmatrix}$ and $x^2 = -l$, then show that $(A + B)^2 = A^2 + B^2$.
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Answer

We have, $\text{A}=\begin{bmatrix}0&-\text{x}\\\text{x}&0\end{bmatrix},\ \text{B}=\begin{bmatrix}0&1\\1&0\end{bmatrix}$ and $x^2 = -1$
$\therefore\ (\text{A}+\text{B})=\begin{bmatrix}0&-\text{x}+1\\\text{x}+1&0\end{bmatrix}$
$\therefore\ (\text{A}+\text{B})^2=\begin{bmatrix}0&-\text{x}+1\\\text{x}+1&0\end{bmatrix}\begin{bmatrix}0&-\text{x}+1\\\text{x}+1&0\end{bmatrix}$ $=\begin{bmatrix}1-\text{x}^2&0\\0&1-\text{x}^2\end{bmatrix}\ ....(\text{i})$ Also, $\text{A}^2=\text{A}.\text{A}=\begin{bmatrix}0&-\text{x}\\\text{x}&0\end{bmatrix}\begin{bmatrix}0&-\text{x}\\\text{x}&0\end{bmatrix}$$=\begin{bmatrix}-\text{x}^2&0\\0&-\text{x}^2\end{bmatrix}$
And $\text{B}^2=\text{B}.\text{B}=\begin{bmatrix}0&1\\1&0\end{bmatrix}\begin{bmatrix}0&1\\1&0\end{bmatrix}$$=\begin{bmatrix}0&1\\1&0\end{bmatrix}$
$\therefore\ \text{A}^2+\text{B}^2=\begin{bmatrix}-\text{x}^2&0\\0&-\text{x}^2\end{bmatrix}+\begin{bmatrix}0&1\\1&0\end{bmatrix}$
$=\begin{bmatrix}1-\text{x}^2&0\\0&1-\text{x}^2\end{bmatrix}\ ....(\text{ii})$
From eq. (i) and (ii), we have $(\text{A}+\text{B})^2=\text{A}^2+\text{B}^2$

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