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  4. If A =[ x 1 [0.3em] 1 0 ] and A2 is the identity matrix, then x is equal to

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Q. If $A =\begin{bmatrix} x & 1 \\[0.3em] 1 & 0 \end{bmatrix}$ and $A^2 $ is the identity matrix, then x is equal to

Matrices

Solution:

Given,$ A = \begin{bmatrix} x & 1 \\[0.3em] 1 & 0 \end{bmatrix} \, \therefore \, A^2 = \begin{bmatrix} x & 1 \\[0.3em] 1 & 0 \end{bmatrix}\begin{bmatrix} x & 1 \\[0.3em] 1 & 0 \end{bmatrix}$
= $\begin{bmatrix} x^2 + 1 & x \\[0.3em] x & 1 \end{bmatrix} =\begin{bmatrix} 1 & 0 \\[0.3em] 0 & 1 \end{bmatrix}$ ($\because \, A^2 $ = I)
$\Rightarrow \, x^2 + 1 = 1 , x = 0 \, \Rightarrow \, x = 0 \,$