How to Find the Inverse of a Matrix?

Finding the inverse of a problem is mostly used to solve the problem more easily in Algebra. For instance, if you’re calculating a division problem, for easy solving you can also use multiplication method with its reciprocal. This method is called an inverse operation. In the case of Matrix, there is no division operator. So, we usually use the opposite process to calculate in the matrix.

Let’s take a 3 X 3 Matrix and find it’s inverse. Finding the inverse of a matrix is a long task. But it’s worth a review. You can see the opposite by creating Adjugate Matrix. Let’s explore them step by step.

In the first step, you need to find the determinant of the matrix. If the result is 0, then you don’t need to proceed ahead as the matrix has no inverse. But if not, then you have to continue with the process. The determinant of the matrix can be represented symbolically as det(M).

If the det(M) value is a non-zero, then let’s move on to the next step. In this step, you have to take the transpose of the original Matrix by transposing means that you’ve to reflect the matrix about the main diagonal, or equivalently, swapping the (i,j)element to (j, i). While doing this process, you’ll see the main diagonal will remain the same as before.

After you got the transpose of the original matrix move on to the next step, the 3 X 3 Transpose Matrix is a composition of the various minor 2 X 2 matrixes. To identify them you have to start by highlighting the row and column of the term you want to begin. The list will include five terms for sure. The rest four will be your first 2 X 2 minor matrix. In this process find all the minor pattern from the first matrix and see their value.

Now replace the results with their respective original matrix. That means the pattern you calculated for the original (1,1) will go on its position. After returning them with their respective positions, you must change the sign of alternating the terms of the new matrix.

In the last step remind again the determinant of the original matrix given to you. Make sure that the determinant is a non-zero value. Now divide every term of the new pattern with the determinant and place each result on its respective places. The new matrix that came out of the procedure will be the Inverse of the original model.