Overview:

# Part 10: Inverses of 3 × 3 matrices

To find the inverse, M-1, of a 3 × 3 matrix M (if M-1 exists), we first need to find the cofactor matrix of M, which is the matrix made up of the 9 cofactors of each element of M. We first came across cofactors in part 9.

We also need to be able to find the transpose of a matrix. We can obtain the transpose of a matrix by writing its rows as its columns and vice versa. This is equivalent to reflecting its elements along its diagonal (from top-left to bottom-right). Here is an example:

If A $$= \begin{pmatrix} 4 & 5 & -7\\ 2 & -3 & 0 \\ 1 & -6 & 8 \\ \end{pmatrix}$$, the the transpose of A, denoted AT is $$\begin{pmatrix} 4 & 2 & 1\\ 5 & -3 & -6 \\ -7 & 0 & 8 \\ \end{pmatrix}$$.

If M has cofactor matrix C and is non-singular, then M-1$$=\frac{1}{\text{det }\textbf{M}}$$CT. Use this applet to practise finding the inverse of 3 × 3 matrices.