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To find the inverse of the matrix \( A \) using the Gauss-Jordan method, we first write the augmented matrix \([A | I]\), where \( I \) is the identity matrix. The matrix \( A \) is: \[ A = \begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & 4 \\ 1 & 2 & 2 \end{bmatrix} \] The identity matrix \( I \) for a 3x3 matrix is: \[ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] Now, augment the matrix \( A \) with the identity matrix: \[ \left[\begin{array}{ccc|ccc} 1 & -1 & 1 & 1 & 0 & 0 \\ 1 & -2 & 4 & 0 & 1 & 0 \\ 1 & 2 & 2 & 0 & 0 & 1 \end{array}\right] \] We will perform row operations to reduce the augmented matrix to the form \([I | A^{-1}]\), where \( A^{-1} \) is the inverse of \( A \). Step 1: Make the element in row 1, ___ _________ _________ _______ _______ ________ ________ ______ ___.
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