Question

If the inverse of the matrix $A=\left[ \begin{matrix}
   -1 & -3 & 3 & -1 \\
   1 & 1 & -1 & 0 \\
   2 & -5 & 2 & -3 \\
   -1 & 1 & 0 & 1 \\
\end{matrix} \right]$ using Gauss Jordan method is ${{A}^{-1}}=\left[ \begin{matrix}
   0 & 2 & 1 & 3 \\
   m & 1 & -1 & -2 \\
   1 & 2 & 0 & 1 \\
   -1 & 1 & 2 & 6 \\
\end{matrix} \right]$, then find the value of m?

Answer 1

Hint: We start solving the problem by using the fact that the multiplication of a matrix and its inverse is the identity matrix. We use the definition of an identity matrix and write-in in the given order of matrices. We then make the operation of multiplication between the two matrices and equate the corresponding elements on both sides to get the value of m.

Complete step-by-step solution:
According to the problem, we are given a matrix $A=\left[ \begin{matrix}
   -1 & -3 & 3 & -1 \\
   1 & 1 & -1 & 0 \\
   2 & -5 & 2 & -3 \\
   -1 & 1 & 0 & 1 \\
\end{matrix} \right]$ and its inverse using gauss Jordan method is ${{A}^{-1}}=\left[ \begin{matrix}
   0 & 2 & 1 & 3 \\
   m & 1 & -1 & -2 \\
   1 & 2 & 0 & 1 \\
   -1 & 1 & 2 & 6 \\
\end{matrix} \right]$. We need to find the value of m.
We know that the multiplication of a matrix and its inverse is equal to the identity matrix.
So, we have $A.{{A}^{-1}}=I$.
We know that the identity matrix I is defined as a square with all the elements of principal diagonal as ‘1’ and all other elements as ‘0’. Since, the order of the matrices given in the problem is $4\times 4$. We need to take the order of the identity matrix as $4\times 4$. So, we get the identity matrix I as $\left[ \begin{matrix}
   1 & 0 & 0 & 0 \\
   0 & 1 & 0 & 0 \\
   0 & 0 & 1 & 0 \\
   0 & 0 & 0 & 1 \\
\end{matrix} \right]$.
Now, we have $\left[ \begin{matrix}
   -1 & -3 & 3 & -1 \\
   1 & 1 & -1 & 0 \\
   2 & -5 & 2 & -3 \\
   -1 & 1 & 0 & 1 \\
\end{matrix} \right]\times \left[ \begin{matrix}
   0 & 2 & 1 & 3 \\
   m & 1 & -1 & -2 \\
   1 & 2 & 0 & 1 \\
   -1 & 1 & 2 & 6 \\
\end{matrix} \right]=\left[ \begin{matrix}
   1 & 0 & 0 & 0 \\
   0 & 1 & 0 & 0 \\
   0 & 0 & 1 & 0 \\
   0 & 0 & 0 & 1 \\
\end{matrix} \right]$.
Let us complete the matrix multiplication and compare the corresponding elements on both sides.
\[\Rightarrow \left[ \begin{matrix}
   \left( 0 \right)+\left( -3m \right)+\left( 3 \right)+\left( 1 \right) & \left( -2 \right)+\left( -3 \right)+\left( 6 \right)+\left( -1 \right) & \left( -1 \right)+\left( 3 \right)+\left( 0 \right)+\left( -2 \right) & \left( -3 \right)+\left( 6 \right)+\left( 3 \right)+\left( -6 \right) \\
   \left( 0 \right)+\left( m \right)+\left( -1 \right)+\left( 0 \right) & \left( 2 \right)+\left( 1 \right)+\left( -2 \right)+\left( 0 \right) & \left( 1 \right)+\left( -1 \right)+\left( 0 \right)+\left( 0 \right) & \left( 3 \right)+\left( -2 \right)+\left( -1 \right)+\left( 0 \right) \\
   \left( 0 \right)+\left( -5m \right)+\left( 2 \right)+\left( 3 \right) & \left( 4 \right)+\left( -5 \right)+\left( 4 \right)+\left( -3 \right) & \left( 2 \right)+\left( 5 \right)+\left( 0 \right)+\left( -6 \right) & \left( 6 \right)+\left( 10 \right)+\left( 2 \right)+\left( -18 \right) \\
   \left( 0 \right)+\left( m \right)+\left( 0 \right)+\left( -1 \right) & \left( -2 \right)+\left( 1 \right)+\left( 0 \right)+\left( 1 \right) & \left( -1 \right)+\left( -1 \right)+\left( 0 \right)+\left( 2 \right) & \left( -3 \right)+\left( -2 \right)+\left( 0 \right)+\left( 6 \right) \\
\end{matrix} \right]=\left[ \begin{matrix}
   1 & 0 & 0 & 0 \\
   0 & 1 & 0 & 0 \\
   0 & 0 & 1 & 0 \\
   0 & 0 & 0 & 1 \\
\end{matrix} \right]\].
\[\Rightarrow \left[ \begin{matrix}
   -3m+4 & 0 & 0 & 0 \\
   m-1 & 1 & 0 & 0 \\
   -5m+5 & 0 & 1 & 0 \\
   m-1 & 0 & 0 & 1 \\
\end{matrix} \right]=\left[ \begin{matrix}
   1 & 0 & 0 & 0 \\
   0 & 1 & 0 & 0 \\
   0 & 0 & 1 & 0 \\
   0 & 0 & 0 & 1 \\
\end{matrix} \right]\].
We know that if two matrices are equal, then the corresponding elements in both matrices are equal.
So, we get $-3m+4=1$.
$\Rightarrow -3m=1-4$.
$\Rightarrow -3m=-3$.
$\Rightarrow m=\dfrac{-3}{-3}$.
$\Rightarrow m=1$.
We have found the value of m as 1.
$\therefore$ The value of m is 1.

Note: We need not always find the inverse of the matrix if both matrix and its inverse are given. We can also verify the value of m by equating the remaining elements in the matrix. We can also find the inverse using the formula ${{A}^{-1}}=\dfrac{AdjA}{DetA}$, but this will take a lot of time calculating for a matrix of order 4. We can also find the value of A by finding the determinant of matrix A and its inverse and using the fact $\det \left( {{A}^{-1}} \right)=\dfrac{1}{\det \left( A \right)}$.