There are two methods for finding inverse of a matrix. Inverse of a matrix can be found by using the identity matrix and also by finding the determinant of the matrix. The method that makes use of identity matrix in finding the inverse of the matrix is called as Gauss-Jordan elimination method. This article on how to calculate inverse of a matrix discusses both these methods.
Inverse of a matrix is usually represented as A-1 i.e. if A is a given matrix then its inverse is represented as A-1. We can find inverse of square matrices only. There are some square matrices that don’t have inverse.
Finding inverse of a matrix by finding determinant:
To find inverse of a matrix using this method you have to find out the determinant of the matrix and use this value in finding the inverse of the matrix. Let me explain this using a simple example.
B =
Steps for finding inverse of this matrix:
Step 1: We have to find out the determinant of the matrix initially.
Therefore,
Determinanat of this matrix = (5 × 2) – (3 × 1)
Determinanat of the matrix = 10 – 3 = 7
Step 2: In the second step we have to swap the elements of the leading diagonal elements. Leading diagonal elements are the elements from top left to the bottom right of the matrix.
Therefore,
B=
Step 3: Now, change the signs of the other diagonal elements as shown below.
Therefore,
B =
Step 4: finally divide each element of the matrix by the determinant of this matrix. This is the inverse of the matrix.
Therefore,
Gauss-Jaurdan elimination method:
This methods makes use of an identity matrix which is of the same order of the matrix whose inverse is to be calculated. Let us use a simple example so that you can easily understand this method. We make use of matrix row operations in this method.
Suppose we have a matrix as,
We have to reduce the matrix whose inverse is to be found out into an identity matrix. The final matrix to which the identity matrix gets transformed when we reduce our matrix to identity matrix is the inverse of our matrix.
Step 2: Now, to reduce the second row in this matrix, multiply each element of first row by -2 and add the obtained values to the corresponding elements of the second row. This reduces the first element of the identity matrix to 0. Same operation should be carried on the identity matrix.
The operation we carried out here can be represented as, R2 = -2R1 + R2
Doing this we have obtained elements the required elements 1 and 0 in the second row.
Step 3: Now, we have to reduce the first. If we multiply second row by -2 and add the elements to the corresponding elements of the first row we get the required values in the first row.
The operation carried out in this step can be represented as,
R1= (-2)R2 + R1
The matrix which we have obtained on the right hand side is the required matrix i.e. it is the inverse of the original matrix.
Therefore,
Note: If a single operation does not give you these values then perform such operations repeatedly until you get the required values.
We can confirm that this matrix is the inverse of the matrix by multiplying the two matrices i.e. the original matrix and the inverse of the matrix. The matrix which is obtained after this multiplication operation is the identity matrix.
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