How to Solve a Set of Equations with Matrices – An Explanatory Guide

Jul 06, 2020

Solving a set of equations through the use of matrices is simply an organized way of applying the elimination method. This method is practically beneficial when the problem involves more than 2 equations comprising more than 2 variables.

The basic idea is pretty easy and can be done through a systematic procedure. This article can be considered a simple guide to that procedure. We have included examples to make the subject matter lucid and easy to understand.

Before we delve deeper into this subject matter, we would presume that you have the knowledge of determinant and matrix; just the basics would do.

Inverse Matrix

Matrices can be used to find an easy solution of a set of equations but before going into that, one must know about the way through which the inverse of a specific Matrix is found out. If a matrix is represented by a letter “C”, the inverse is represented by “C-1”. Let’s go through the rule now then.

A matrix C can have an inverse (C-1) if and only if the determinant of the matrix C is NOT equal to 0. By that, we mean -

Now, let’s go to the main process. The image depicted below will provide you with a detailed idea. We have included one example as well for your reference.

Solving Equations through Matrices

We have already told you before that solution of equations through matrices is an easy as well as an organized method of implementing the technique of “elimination”.

Consider a problem where you are provided with 3 set of equations. For example -

Those equations can be turned into a table consisting of numbers like -

So we have basically represented the set of coefficients on both sides of the equation and have eliminated the variables. However, that is not the correct matrix representation of the equations. The correct one should be -


Easy enough, right?

Now let’s get back to our problem. Remember that you have to find out the values of x, y, z. The matrix solution can be done through the process -

or, X = A-1B.
X= The matrix having the variables x, y, z.
A= The matrix having the coefficients of x, y, z.
B= The numbers on the right hand side i.e. the respective solutions of the equation.

Substituting the matrices respectively, we get -

At first, it’s necessary to find out the value of A-1. Keep in mind that it’s a 3x3 matrix and hence, a little tricky. By tricky, we mean lengthy but we can guarantee that it’s not difficult like your other topic involving calculus, coordinate geometry, trigonometry etc. Pay special attention now.

So we have determined the value of A-1. Now it’s time we do the main solution. This part is the easiest of the lot. The main trick was to find the inverse matrix that requires considerable hard work and concentration. Now let’s finish our problem.

So, that’s done and dusted and we have found out the required values of x, y, and z. The process is pretty methodical and seemingly sequential you just have to follow one sequence after another to come to a solution. The main trick is to determine the adjoint matrix.

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Article Posted in: How to


Sid writes educational content periodically for Wizert and backs it up with extensive research and relevant examples. He's an avid reader and a tech enthusiast at the same time with a little bit of “Arsenal Football Club” thrown in as well. He's got more than 5 years of experience in technical content framing, digital marketing, SEO and graphic designing.

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