Method 2 (an example of dinosaur mathematics - should be extinct)

The inverse of a 3×3 matrix is given by:

"adj A" is short for "the adjoint of A". We use cofactors (that we met earlier) to determine the adjoint of a matrix.

Cofactors

Recall: The cofactor of an element in a matrix is the value obtained by evaluating the determinant formed by the elements not in that particular row or column.

Example: Consider the matrix

The cofactor of 6 is

The cofactor of -3 is

We find the adjoint matrix by replacing each element in the matrix with its cofactor and applying a + or - sign as follows:

and then finding the transpose of the resulting matrix. The transpose means the 1^st column becomes the 1^st row; 2^nd column becomes 2^nd row, etc.

Example 1:

Find the inverse of the following by using the adjoint matrix method:

A =

Solution:

Step 1:

Replace elements with cofactors and apply + and -

Step 2

Transpose the matrix:

adjA =

Before we can find the inverse of matrix A, we need det A:

Now we have what we need to apply the formula

So

A^-1 =

Example 2:

Find the inverse of

using Method 2.

Answer:

Interchange rows and columns:

Det A =

So

Now let's see how to do all this more appropriately using a computer...