Definition:Simultaneous Linear Equations/Matrix Representation/Augmented Matrix
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Definition
Consider the system of simultaneous linear equations can be expressed as:
- $\ds \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
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.
Thus:
- $\begin {bmatrix} \mathbf A & \mathbf b \end {bmatrix} = \begin {bmatrix}
\alpha_{1 1} & \alpha_{1 2} & \cdots & \alpha_{1 n} & \beta_1 \\ \alpha_{2 1} & \alpha_{2 2} & \cdots & \alpha_{2 n} & \beta_2 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
\alpha_{m 1} & \alpha_{m 2} & \cdots & \alpha_{m n} & \beta_m \\ \end {bmatrix}$
Also see
Sources
- 1982: A.O. Morris: Linear Algebra: An Introduction (2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): augmented matrix
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): augmented matrix
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): augmented matrix