Linear Algebra/Augmented Matrices

The Augmented matrix is a Matrix that is constructed by combining both the Coefficient matrix and a vector which might represent the solution for the system of equations. Or can be constructed by combining two matrices. But an important rule must be taken in consideration the two matrices must have the same number of rows. For example if we have a matrix of size (m x n) matrix we should have another matrix of size (m x k) where n is a positive natural number.

$$ A_{(m x n)} = \begin{bmatrix} a11 & a12 &. & . & . & a1n \\ a21 & a22 &. & . & . & a2n \\. & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ am1 & am2 &. & . & . & amn \end{bmatrix} $$

$$ B_{(m x k)} = \begin{bmatrix} b11 & b12 &. & . & . & b1n \\ b21 & b22 &. & . & . & b2n \\. & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ bm1 & bm2 &. & . & . & bmn \end{bmatrix} $$

The Augmented matrix (A|B) in this case will be:

$$ (A|B) = \begin{bmatrix} a11 & a12 &. & . & . & a1n \\ a21 & a22 &. & . & . & a2n \\. & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ am1 & am2 &. & . & . & amn \end{bmatrix} \quad \Bigg| \quad \begin{bmatrix}  b11 & b12 &. & . & . & b1n \\ b21 & b22 &. & . & . & b2n \\. & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ bm1 & bm2 &. & . & . & bmn \end{bmatrix}$$

Also if we have a system of equations we can express it as an augmented matrix which makes it easier to work with. And, this becomes very useful when we have systems that have relatively large dimensions, for Example:

$$2x + 3y + 4z = 9\,$$ $$3x + 5y + 8z = 21\,$$ $$4x + 9y + z = 17\,$$

Can Be Expressed as the following augmented matrix: $$\begin{bmatrix} 2 & 3 & 4 \\ 3 & 5 & 8 \\ 4 & 9 & 1 \end{bmatrix} \Bigg| \begin{bmatrix} 9 \\ 21 \\ 17  \end{bmatrix}$$ Finally, the augmented matrix is very useful when it comes to solving systems, finding the inverse of a matrix, etc...