Inverse Matrices

Question

Find the inverse of the following matrices if it exists:

1.

[ 1 -2 ]

[ -3 5 ]

2.

[ -2 0 2 ]

[ 4 -2 -1 ]

[ 0 2 -2 ]

Answer 1

Write augmented matrix

[ 1 -2 | 1 0 ]

[ -3 5 | 0 1 ]

add 3x first row to second row.

[ 1 -2 | 1 0 ]

[ 0 -1 | 3 1 ]

multiply second row by -1

[ 1 -2 | 1 0 ]

[ 0 1 | -3 -1 ]

then add 2x second row to first row.

[ 1 0 | -5 -2 ]

[ 0 1 | -3 -1 ]

Inverse is

[ -5 -2 ]

[ -3 -1 ]

You can check using matrix multiplication.

For the 3x3 matrix, it's laborious to type, but the idea is the same: augment with the identity matrix on the right. Perform row operations until you have the identity matrix on the left. At that point, the new matrix on the right is the inverse of the one you started with. If you cannot get to the identity matrix on the left with row operations, then the matrix you started with is not invertible.