Math 2.0

Linear and multilinear algebra; matrix theory Definition

Augmented matrix

Builds on Coefficient matrix

Definition

Let FF be a field. Take a system of linear equations Ax=bAx = b over FF, with coefficient matrix AA of size m×nm \times n and constant vector b∈Fmb \in F^m. The augmented matrix of the system is the m×(n+1)m \times (n+1) matrix [A∣b][A \mid b]. Its first nn columns are the columns of AA, and its last column is bb.

Remarks

The system 2x+y=52x + y = 5, x−3y=0x - 3y = 0 has the augmented matrix (2151−30)\left(\begin{array}{cc|c} 2 & 1 & 5 \\ 1 & -3 & 0 \end{array}\right). Elementary row operations on the augmented matrix do not change the solution set of the system. More generally, [A∣B][A \mid B] denotes the matrix AA with the columns of a matrix BB added on the right.

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