Definition of augmented matrix (of a linear system)
Created almost 9 years ago, updated 5 months ago
ID23755
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):
**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.