\[ M=\left[\ \begin{matrix}7&x\\-8&-7\\\end{matrix}\ \right]\]
The aim of the article is to find the value of the variable $x$ within the given matrix for which it will be equal to its inverse matrix.
The basic concept behind this question is the understanding of the Matrix, how to find the determinant of a matrix, and the inverse of a matrix.
For a matrix $A$, the inverse of its matrix is represented by the following formula:
\[A^{ -1} = \dfrac{1}{det\space A} Adj\ A\]
Where:
$A^{ -1} = inverse \space of \space matrix$
$det\space A = Determinant \space of \space matrix$
$Adj\ A= Adjoint \space of \space matrix$
Expert Answer
Let us suppose the given matrix is $M$:
\[ M=\left[\ \begin{matrix}7&x\\-8&-7\\\end{matrix}\ \right]\]
For the given condition in the question, we know that the matrix should be equal to its inverse so we can write it as follows:
\[M = M^{-1 }\]
We know that the inverse of a matrix is determined by the following formula:
\[M^{ -1} = \dfrac{1}{det\space M} Adj\ M\]
Now first to find out the determinant of matrix $M$:
\[ det\ M = 7(-7) -x (-8)\]
\[ det\ M = -49 +8x \]
\[ det\ M = 8x -49 \]
Now we will find the Adjoint of the matrix $M$ as follows:
\[ M=\left[\ \begin{matrix}7&x\\-8&-7\\\end{matrix}\ \right] \]
\[ Adj\ M\ = \left[\ \begin{matrix} -7&-x\\8&7\\\end{matrix}\ \right] \]
To find the inverse of the matrix, we will put the values of its determinant and adjoint in the following formula:
\[M^{ -1} = \dfrac{1}{det\space M} Adj\ M\]
\[M^{ -1} = \dfrac{1}{8x -49} \times \left[\ \begin{matrix} -7&-x\\8&7\\\end{matrix}\ \right] \]
\[M^{ -1} = \left[\ \begin{matrix}\dfrac{-7}{8x -49} &\dfrac{-x}{8x -49}\\\dfrac{8}{8x -49}&\dfrac{7}{8x -49}\\\end{matrix}\ \right] \]
According to the condition given in the question, we have:
\[M = M^{-1 }\]
Putting the matrix $M$ and its inverse here, we have:
\[ \left[\ \begin{matrix}7&x\\-8&-7\\\end{matrix}\ \right] = \left[\ \begin{matrix}\dfrac{-7}{8x -49} &\dfrac{-x}{8x -49}\\\dfrac{8}{8x -49}&\dfrac{7}{8x -49}\\\end{matrix}\ \right] \]
Now compare the matrices on both sides so that we can find out the value of $x$. For this put any of the four equations equal to the equation in the other matrix in the same position. We have chosen the first equation, so we get:
\[ 7 = \dfrac{-7}{8x-49} \]
\[ 7 (8x-49) = -7 \]
\[ 56x-343 = -7 \]
\[ 56x = 343 -7 \]
\[ 56x = 336 \]
\[ x = \dfrac {336}{56} \]
\[ x = 6 \]
So the value of $x$ for which the matrix will be equal to its inverse is $x=6$.
Numerical Results
For the given matrix $\left[\ \begin{matrix}7&x\\-8&-7\\\end{matrix}\ \right]$ it will be equal to its inverse when the value of $x$ will be:
\[ x = 6 \]
Example
For the given matrix $\left[\ \begin{matrix}2&x\\-8&-2\\\end{matrix}\ \right]$ find the determinant and adjoint.
Solution
Let us suppose the given matrix is $Y$:
\[Y=\left[\ \begin{matrix}2&x\\-8&-2\\\end{matrix}\ \right]\]
Now first to find out the determinant of matrix $Y$:
\[det\ Y=2(-2) -x (-8)\]
\[det\ Y=-4 +8x\]
\[det\ Y=8x -4\]
Adjoint of the matrix $Y$:
\[Y=\left[ \begin{matrix}2&x\\-8&-2\\\end{matrix}\ \right]\]
\[Adj\ Y=\left[ \begin{matrix} -2&-x\\8&2\\\end{matrix}\ \right]\]