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To understand the Gauss-Jordan method, we first need to know what a matrix inverse is. The inverse of a matrix \(\textbf{A}\) is another matrix, denoted as \(\textbf{A}^{-1}\), such that when you multiply them together, you get the identity matrix \(\textbf{I}\). The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. For a matrix to have an inverse, it must be square (same number of rows and columns) and its determinant must be non-zero. The equation looks like this: \[ \textbf{A} \textbf{A}^{-1} = \textbf{I} \] In this exercise, we use the Gauss-Jordan method to find \(\textbf{A}^{-1}\).

Row operations are transformations we perform on the rows of a matrix to simplify it. There are three main types:

• Swapping two rows
• Multiplying a row by a non-zero scalar
• Adding or subtracting the multiple of one row to/from another row

These operations help us to systematically reduce the matrix. In the case of finding an inverse, we perform these operations to turn the original matrix into the identity matrix while applying the same operations to another matrix simultaneously. The steps in the example involve:

• Dividing the first row by 3
• Using the first row to eliminate the first column in the second row
• Dividing the second row by \(\frac{4}{3}\)
• Using the second row to eliminate the second column in the first row

By the end, the left side of our augmented matrix becomes the identity matrix, and the right side becomes the inverse.

An augmented matrix is a combination of two matrices. In this case, we combine the original matrix \(\textbf{A}\) with the identity matrix \(\textbf{I}\). The augmented matrix looks like this:
\[ \begin{pmatrix} 3 & 2 & | & 1 & 0 \ 5 & 3 & | & 0 & 1 \ \end{pmatrix} \] This setup allows us to apply row operations simultaneously to both \(\textbf{A}\) and \(\textbf{I}\) in order to transform the identity matrix into the inverse \(\textbf{A}^{-1}\). As we perform row operations to get \(\textbf{A}\) into its reduced row echelon form, we also transform \(\textbf{I}\) into \(\textbf{A}^{-1}\). This method ensures that when \(\textbf{A}\) becomes \(\textbf{I}\), the augmented identity matrix becomes the inverse of \(\textbf{A}\) on the right side.