Gauss-Jordan Elimination solve Inverse Matrix

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

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

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

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:

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

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

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

     This gives:

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

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

     $$R_1 \leftarrow R_1 - R_2$$

     This gives:

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

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:

$$A^{-1} = \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2} \end{pmatrix}$$

Thus, the inverse of the matrix $A = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ is $\begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2} \end{pmatrix}$.