Definition:Augmented Matrix

Definition

Let $\mathbf A$ be a matrix of order $n \times m$.

Let $\mathbf B$ be a matrix of order $n \times k$.

The augmented matrix of $\mathbf A$ and $\mathbf B$ is the block matrix $\begin {pmatrix} \mathbf A & \mathbf B \end {pmatrix}$ of order $n \times \paren {m + k}$.

Augmented Matrix of Simultaneous Equations

Consider the system of simultaneous linear equations:

$\ds \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$

expressed in matrix representation as:

$\mathbf A \mathbf x = \mathbf b$

Let $\begin {bmatrix} \mathbf A & \mathbf b \end {bmatrix}$ be the block matrix formed from $\mathbf A$ and $\mathbf b$.

Then $\begin {bmatrix} \mathbf A & \mathbf b \end {bmatrix}$ is known as the augmented matrix of the system.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): augmented matrix
• 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations