Gauss-Jordan Elimination solve Inverse Matrix

To find the inverse of the matrix $A=\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$ using Gauss-Jordan elimination, we can follow these steps:

1. Form the augmented matrix by appending the identity matrix to $A$:

   $$\left(\begin{array}{cccc}1& 1& 1& 0\\ 1& -1& 0& 1\end{array}\right)$$

2. Apply row operations to transform the left part of the augmented matrix into the identity matrix.

   • Step 1: Subtract the first row from the second row:

     $$R_2\leftarrow R_2-R_1$$

     This gives:

     $$\left(\begin{array}{cccc}1& 1& 1& 0\\ 0& -2& -1& 1\end{array}\right)$$

   • Step 2: Divide the second row by $-2$:

     $$R_2\leftarrow \frac{R_2}{-2}$$

     This gives:

     $$\left(\begin{array}{cccc}1& 1& 1& 0\\ 0& 1& \frac{1}{2}& -\frac{1}{2}\end{array}\right)$$

   • Step 3: Subtract the second row from the first row:

     $$R_1\leftarrow R_1-R_2$$

     This gives:

     $$\left(\begin{array}{cccc}1& 0& \frac{1}{2}& \frac{1}{2}\\ 0& 1& \frac{1}{2}& -\frac{1}{2}\end{array}\right)$$

The left part of the augmented matrix is now the identity matrix, and the right part is the inverse of $A$.

Therefore, the inverse matrix $A^{-1}$ is:

$$A^{-1}=\left(\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& -\frac{1}{2}\end{array}\right)$$

Thus, the inverse of the matrix $A=\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$ is $\left(\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& -\frac{1}{2}\end{array}\right)$.