Short cut for 2x2 matrices

There is a short cut for finding the inverse of a 2 by 2 matrix. This short cut involves the use of a number called the determinant of the matrix.

For a 2 by matrix A of the form

the determinant of A , which is abreviated by det( A ), is

det( A ) = ad - bc

(Note we sometimes say that det( A ) is the product of the two numbers on the main diagonal minus the product of the two numbers on the other diagonal.)

Here are some 2 by 2 matrices and their determinants.

Example 1

Example 2

Example 3

One application of the determinant is that its value tells us whether or not a matrix has an inverse. We've seen matrix B in the previous section on finding inverses and saw that it has no inverse. When the determinant is zero, the matrix does not have an inverse.

The reason for this can be seen in the formula for the inverse of a 2 by 2 matrix.

Short Cut

The inverse of the matrix

is given by

provided that

Note that the entries on the main diagonal are switched, the signs are changed on the other entries, and the entire matrix is multiplied by the reciprocal of the determinant.

Find the inverses for examples one and three. (Note that in example two the determinant was zero and hence there is no inverse.)

In example one,

This had a determinant of 1, thus its inverse is

In example three,

This had a determinant of 62. Its inverse is