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Augmented Matrices

Coefficient Matrix

The matrix derived from the coefficients of the system of linear equations , not including the constant terms is the coefficient matrix of the system. Make sure, each equation written in standard form with the constant term on the right.

$\begin{array}{cc}\text{system}& \{\begin{array}{l}x+4y+3z=12\hfill \\ 2x-3y-z=20\hfill \\ 4x+5y=15\hfill \end{array}\\ \text{coefficient matrix}& \left[\begin{array}{rrr}1& 4& 3\\ 2& -3& -1\\ 4& 5& 0\end{array}\right]\end{array}$

Note the use of $0$ for the missing $z$ -variable in the third equation.

Augmented Matrix

A matrix derived from a system of linear equations is the augmented matrix of the system. Make sure, each equation written in standard form with the constant term on the right.

$\begin{array}{cc}\text{system}& \{\begin{array}{l}x+4y+3z=12\hfill \\ 2x-3y-z=20\hfill \\ 4x+5y=15\hfill \end{array}\\ \text{coefficient matrix}& \left[\begin{array}{rrrr}1& 4& 3& 12\\ 2& -3& -1& 20\\ 4& 5& 0& 15\end{array}\right]\end{array}$

The first three columns of the augmented matrix show the coefficients of $x$ , $y$ , and $z$ in the linear system. The fourth column in the augmented matrix shows the constant terms in the linear system. The optional dotted line helps to identify the constant term.

When forming either the coefficient matrix or augmented matrix of a system, begin by vertically aligning the variables in the equations and using $0$ 's for the missing variables.