Overview:

1. Introduction to matrices
2. Adding and subtracting matrices
3. Multiplying matrices
4. 2 × 2 Matrices and linear transformations
5. Determinants of 2 × 2 matrices
6. Inverses of 2 × 2 matrices
7. Invariant points and lines in 2 dimensions
8. 3 × 3 Matrices and linear transformations
9. Determinants of 3 × 3 matrices
10. Inverses of 3 × 3 matrices
11. Matrices and simultaneous equations

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 A^T 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\(=\frac{1}{\text{det }\textbf{M}}\)C^T. Use this applet to practise finding the inverse of 3 × 3 matrices.