# How to Solve a System of Equations Using Matrices?

Any system of linear equations can be written as a matrix equation. In this step-by-step guide, you learn how to solve a system of equations using matrices.

**Related Topics**

**A step-by-step guide to solving a system of equations using matrices**

When solving a system of equations using matrices, we have three matrices \(A, B\), and \(X\), where \(A\) is known as the coefficient matrix, \(B\) is the constant matrix, and \(X\) contains all the variables of the equations, known as a variable matrix. Matrix \(A\) is of the order \(m×n\), while \(B\) is the column matrix of the order \(m×1\). The product of matrix \(A\) and matrix \(X\) reaches matrix \(B\). Hence, \(X\) is a column matrix of order \(n×1\).

The matrices are arranged as:

\(\color{blue}{A . X = B}\)

Let’s understand how to solve a system of equations using a matrix with the help of an example. We have a set of two equations given below. The equations are:

\(\begin{cases}x+y=6 \\ 2x+3y=14\end{cases}\)

Arrange all the coefficients, variables, and constants of the matrix in such a way that whenever we find the product of the matrices, the obtained result should be an equation. Then the matrix equation is, \(AX = B\) where:

\(A= \begin{bmatrix}1 & 1 \\2 & 3 \end{bmatrix}\)

\(X= \begin{bmatrix}x \\y \end{bmatrix}\)

\(B= \begin{bmatrix}6 \\14 \end{bmatrix}\)

We need to find matrix \(X\), to solve the equations. It can be found by multiplying the inverse of matrix \(A\) with \(B\), which is obtained as \(X=\left(A^{-1}\right)B\).

To find the determinant of matrix \(A\), we follow the following steps:

\(|A|= \begin{bmatrix}1 & 1 \\2 & 3 \end{bmatrix}\)

Therefore, \(|A|= 3\: – 2 = 1\)

\(|A|≠0\), it is possible to find the inverse of matrix \(A\).

Now, by using the formula for finding the inverse of \(2×2\) matrix:

\(A^{-1}= \begin{bmatrix}3 & -1 \\-2 & 1 \end{bmatrix}\)

Now to find the matrix \(X\), we’ll multiply \(A^{-1}\) and \(B\). We get,

\(\begin{bmatrix}3 & -1 \\-2 & 1 \end{bmatrix}\)\(\begin{bmatrix}6 \\14 \end{bmatrix}\)\(=\begin{bmatrix}4\\2 \end{bmatrix}\)

So, the value of matrix \(X\) is, \(\begin{bmatrix}4\\2 \end{bmatrix}\).

## Related to This Article

### More math articles

- HSPT Math Formulas
- ALEKS Math FREE Sample Practice Questions
- A Deep Dive into the Integral Test
- Top 10 Tips You MUST Know to Retake the TSI Math
- Digital Tools for Teaching Math at a Distance
- GED Math Practice Test Questions
- How to Solve Word Problems by Adding Three or More Decimals
- Top 10 SAT Math Practice Questions
- A Comprehensive Collection of Free Praxis Math Practice Tests
- Entertain Your Child Indoors with These Fun, Educational Activities

## What people say about "How to Solve a System of Equations Using Matrices? - Effortless Math: We Help Students Learn to LOVE Mathematics"?

No one replied yet.