AcademyLinear Systems
Academy
Augmented Matrices
Level 1 - Linear Algebra topic page in Linear Systems.
Principle
An augmented matrix records a linear system without rewriting the variables every time. The coefficient matrix goes on the left, the right-hand side goes in the final column, and a vertical bar separates the two parts.
Augmented matrices are bookkeeping devices for solving systems by row operations.
Notation
\(A\)
the coefficient matrix of a linear system
\(\mathbf x\)
the column vector of unknowns
\(\mathbf b\)
the right-hand side vector
\(A\mathbf x=\mathbf b\)
matrix form of a linear system
\([A\mid\mathbf b]\)
the augmented matrix of the system
\(a_{ij}\)
coefficient of unknown j in equation i
The Core Method
For a system
System form
\[A\mathbf x=\mathbf b\]
write the augmented matrix as
Augmented matrix
\[[A\mid\mathbf b]\]
The left block stores the coefficients, and the right column stores the constants. Each row is one equation.
When converting between a system and an augmented matrix, keep the variable order fixed. If the variable order is \(x,y,z\), then the columns of \(A\) must always mean coefficients of \(x\), then \(y\), then \(z\).
Worked Cases
Question
Write the augmented matrix for \(2x-y+3z=5\), \(-x+4y=7\), and \(3x+z=-2\), using variable order \(x,y,z\).
Answer
Read coefficients in the order \(x,y,z\). The first row is \(2,-1,3\mid5\). The second equation has no \(z\)-term, so its row is \(-1,4,0\mid7\). The third equation has no \(y\)-term, so its row is \(3,0,1\mid-2\). Therefore the augmented matrix is \(\begin{pmatrix}2&-1&3&|&5\\-1&4&0&|&7\\3&0&1&|&-2\end{pmatrix}\).
Examples
Question
Why must a missing variable be recorded with coefficient \(0\)?
Answer
The columns must keep a fixed variable order. If \(z\) is missing from \(2x-y=3\), its coefficient is \(0\), so the row for variables \(x,y,z\) is \(2,-1,0\mid3\). Omitting the zero would shift columns and change the meaning of later entries.