Find x such that the matrix is equal to its own inverse. A=begin{bmatrix}7 & x -8 & -7 end{bmatrix}

Step 1
Given matrix,
$A=\left[\begin{array}{cc}7& x\\ -8& -7\end{array}\right]$
And,
$A=A^{-1}$
Formula for an inverse of a matrix is
$A^{-1}=\frac{1}{detA}adjA$
Step 2 Now, $det(A)=7(-7)-x(-8)$
$det(A)=-49+8x$
$det(A)=8x-49$
And,
$adjA=\left[\begin{array}{cc}-7& -x\\ 8& 7\end{array}\right]$
so,
$A^{-1}=\left[\begin{array}{cc}-\frac{7}{8x-49}& -\frac{x}{8x-49}\\ \frac{8}{8x-49}& \frac{7}{8x-49}\end{array}\right]$
Step 3
Now , as given
$A=A^{-1}$
$\left[\begin{array}{cc}7& x\\ -8& -7\end{array}\right]=\left[\begin{array}{cc}-\frac{7}{8x-49}& -\frac{x}{8x-49}\\ \frac{8}{8x-49}& \frac{7}{8x-49}\end{array}\right]$
on comparing the matrices,
$7=\frac{7}{8x-49}$
$7(8x-49)=-7$
$56x-343=-7$
$56x=343-7$
$56x=343-7$
$56x=336$
$x=\frac{336}{56}$
x=6
Step 4
Therefore,
The value of x such that the matrix is equal to its own inverse is
x=6