Solving Systems of Linear Equations with Matrices

In an earlier article I solved a 3x3 system by elimination; here is the same system written with matrices and solved by row reduction on the augmented matrix. $$ \begin{cases} x + y + z = 6 \\ 2x - y + z = 3 \\ 3x + 2y - z = 4 \end{cases} $$The same information fits into one matrix equation \(A\mathbf{x} = \mathbf{b}\), where $$ A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & -1 & 1 \\ 3 & 2 & -1 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 6 \\ 3 \\ 4 \end{pmatrix}. $$Each row of \(A\) holds the coefficients of one equation, and \(\mathbf{b}\) holds the constants on the right-hand side. Augmented matrix We pack \(A\) and \(\mathbf{b}\) into the augmented matrix \([A \mid \mathbf{b}]\). Row operations on this matrix are exactly the legitimate moves from elimination: add a multiple of one row to another, multiply a row by a nonzero scalar, swap two rows. $$ \left(\begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 2 & -1 & 1 & 3 \\ 3 & 2 & -1 &…