Part 7 : Inverses and Gauss-Jordan Elimination

Inverse of a matrix

Assuming a matrix A.

The inverse of matrix A will be another matrix of same order, which on multiplication with A return I (Identity Matrix) as product.

Inverse of A is denoted with letter representing matrix raised to power (-1).

Inverse of Matrix A

Such that

Identity matrix has same order

Multiplication of matrices is not commutative but when we multiply a matrix with its inverse multiplication can be commutative.

Order doesn’t matter when a matrix is multiplied with its inverse

Also, only square matrices have inverses.

Matrix Division

Division of matrices was not discussed in part 2.

Now, with the help of inverses we can perform division of two matrices.

Taking an example

This is equivalent to multiplication of matrix B with inverse of A.

As

Then

Resulting matrix

Gauss-Jordan Elimination

An algorithm to find inverse of a given matrix, it is similar to Gaussian elimination or we can say it is Gaussian elimination extended to one more step.

It is named after Carl Friedrich Gauss and Wilhelm Jordan, a German geodesist .

Wilhelm Jordan improved the stability of algorithm.

Carl Friedrich Gauss

Wilhelm Jordan

To perform Gauss-Jordan Elimination we have to :

1. Make augmented matrix from given matrix and its identity matrix (Order of Identity matrix is decided according to the order of given matrix).

2. Perform elimination (as in step 2 of Gaussian elimination), aiming to obtain row echelon form on left half of augmented matrix.

3. Reduce it further to get Reduced Row Echelon Form (Identity matrix) on left half of augmented matrix.

4.The right half of augmented matrix, is the inverse of given matrix.

Assuming that we have to find inverse of matrix A (above) through Gauss-Jordan Elimination.

Step 1 (Make Augmented matrix) :

Matrix A

Identity matrix for matrix A

Augmented matrix of matrix A and I

Step 2 (Elimination) :

Applying the operations

Elements eliminated from first column

Now, we apply similar operations for column 2.

Row echelon form

If we were doing Gaussian Elimination, we would’ve finished by now and started back substitution but to find inverse we have to take it one step further. We have to find Reduced Row Echelon Form

Step 3 (Elimination to Reduced Row Echelon Form) :

For obtaining reduced row echelon form, we aim to convert left half of augmented matrix into an identity matrix, by using similar operations as in Gaussian Elimination and we start from bottom row to top.

First, we reduce row 3 to an identity matrix form.

Then, we reduce row 2.

Second row is reduced

Now, we reduce row 1 and we will get reduced row echelon form.

Reduced Row Echelon Form

Step 4 (Separate right half of augmented matrix) :

The right half of augmented matrix is the inverse of matrix A.

Inverse of matrix A

Are all matrices invertible?

No, a matrix should satisfy the following two conditions to be invertible.

1. It should be a square matrix.

2. It should not be a singular matrix (its determinant is not equal to 0).

Read Part 8 : Linear Independence, Rank of Matrix, and Span

You can view the complete series here
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