Definition of augmented matrix (of a linear system)

Definition: Augmented Matrix

The augmented matrix of a linear system is formed by appending the constant vector (right-hand side) as the last column of the coefficient matrix, separated by a vertical bar for clarity.

For the system Ax = b where A is m × n, the augmented matrix is m × (n+1):

[a₁₁ a₁₂ ... a₁ₙ | b₁]
[a₂₁ a₂₂ ... a₂ₙ | b₂]
[...  ...  ... ... | ...]
[aₘ₁ aₘ₂ ... aₘₙ | bₘ]

Example: For the system:

2x + y - z = 8
-3x - y + 2z = -11
-2x + y + 2z = -3

The augmented matrix is:

[ 2  1 -1 |  8]
[-3 -1  2 |-11]
[-2  1  2 | -3]

The augmented matrix contains all the information needed to solve the system using row reduction.