Matrices can be used to compactly write and work with systems of equations. As we have learned in previous sections, matrices can be manipulated in any way that a normal equation can be. This is very helpful when we start to work with systems of equations. It is helpful to understand how to organize matrices to solve these systems.

Writing a System of Equations with Matrices

It is possible to solve this system using the elimination or substitution method, but it is also possible to do it with a matrix operation. Before we start setting up the matrices, it is important to do the following:

• Make sure that all of the equations are written in a similar manner, meaning the variables need to all be in the same order.
• Make sure that one side of the equation is only variables and their coefficients, and the other side is just constants.

Solving a system of linear equations using the inverse of a matrix requires the definition of two new matrices: $X$ is the matrix representing the variables of the system, and $B$ is the matrix representing the constants. Using matrix multiplication, we may define a system of equations with the same number of equations as variables as:

$\displaystyle A\cdot X=B$

To solve a system of linear equations using an inverse matrix, let $A$ be the coefficient matrix, let $X$ be the variable matrix, and let $B$ be the constant matrix.

Given the system:  

$\displaystyle \begin{aligned} x+8y&=7 \\ 2x-8y&=-3 \end{aligned}$

The coefficient matrix is:

$A=\begin{bmatrix} 1 & 8\\ 2 & -8 \end{bmatrix}$

The variable matrix is:

$\displaystyle X=\begin{bmatrix} x\\ y \end{bmatrix}$

The constant matrix is:

$\displaystyle B=\begin{bmatrix} 7\\ -3 \end{bmatrix}$

Thus, to solve a system $AX=B$, for $X$, multiply both sides by the inverse of $A$ and we shall obtain the solution:

$\displaystyle X=(A^{-1})B$

Provided the inverse $\left( A^{-1} \right)$ exists, this formula will solve the system.

If the coefficient matrix is not invertible, the system could be inconsistent and have no solution, or be dependent and have infinitely many solutions.