Finding the Inverses of 3x3 Matrices

Finding the inverse of a 3x3 matrix is quite involved. The steps are:

1. Form the adjoint matrix.

2. Permute the signs according to

3. Take the transpose.

4. Find the determinantof the original matrix and divide the matrix by it, which means multiplying the matrix by a factor

Example: Find the inverse of

We find the adjoint matrix by, for each element, crossing out the elements in the same row and column, and finding the determinant of the submatrix left behind. For instannce, take the element 3 in the top left hand corner. Cross out the top row and the first column. The submatrix left behind iswith determinant 0*1-2*2=-4. This goes in place of the 3. The adjoint matrix formed in this way is:

Now permute the signs, which results in those signs labelled with a “–“ in 2. above changing signs.

Take the transpose to obtain

Finally multiply the matrix by the reciprocal of the determinant of the original matrix.

The inverse is