invmat.html

Finding the inverse

To find the inverse of an n by n matrix A

1. Form the augmented n by 2n matrix [ A I ] where I is the n by n identity matrix

2. Use elementary row operations to try to transform this into the form [ I B ]. If this can be done then B is the inverse of A.

Example 1

Find the inverse of

Solution .

Form the augmented matrix

Since we need a 1 in the first row and first column, we can get it easily by switching rows.

Switch row 1 and row 2

To have the identity appear in the first two columns we must get a 0 below the 1 in the first column. To do this

Add to row 2 -3 times row 1

We need a 1 in the second row and second column. To get this

Multiply row 2 by -1

To get the last 0 that we need

Add to row 1 -2 times row 2

Since we have gotten a matrix of the form [ I B ] the inverse of A is B. The inverse is

Example 2

Find the inverse of

Solution .

The augmented matrix is

Multiply row 1 by 1/3

Add to row 2 -6 times row 1

The two zeros in the first two columns of the last row indicate that the corresponding system of equations that we are solving to find the inverse has no solution. Thus, the matrix we started with has no inverse.

Example 3

Find the inverse of

Solution .

The augmented matrix given below has 3 rows and 6 columns. Our goal is to use the row operations to transform this into an augmented matrix which has the identity on the left and the inverse on the right.