Chapter 1: Finding the Inverse of a 3×3 Matrix and Solving a System of Linear Equations

Matrices are convenient tools that come in handy in various situations, from organizing large amounts of data to predicting long term trends. Even though matrices are utilized mostly in complex situations, there are also simpler ways to apply them. One unique and simple way to apply matrices is to solve linear equations with the aid of matrices. One might think that solving linear equations is simple, but from what I have seen many students tend to forget how to apply substitution or elimination to solve these types of equations.

As shown above, we are able to solve a system of linear equations using matrices. Although this method requires a lot of steps, it can also be used to solve linear equations with more than three variables. Hence, there should be quite a few situations where using matrices to solve linear equations may be an efficient approach compared to substitution or elimination. Furthermore, as you may have noticed, the number of rows and columns are the same as the number of variables. In the case shown above, there are three variables; therefore, there are three columns or rows (columns and rows for matrix A) in each matrix. Finding the inverse plays a huge part when solving linear equations using matrices. In class, we learned how to find the inverse of a 2×2 matrix, but the steps shown below need to be performed in order to find the inverse of a 3×3 matrix.

Although there are a couple of ways to find the inverse of a 3×3 matrix, this was the most understandable approach. The cofactor matrix, a type of matrix used to calculate the inverse and the determinant, has to be found in order to determine the adjugate matrix. This is because the adjugate matrix is the transpose of the cofactor matrix. Even though we have not explored these concepts in class, it is still possible to find the inverse of the 3×3 matrix as long as we understand the steps required to do so. Finding the inverse of a matrix with more than three rows and columns follows the same rule but is more complicated and requires additional steps; therefore, the use of calculator is strongly suggested for those types of situations.