- Introduction to matrices
- Adding and subtracting matrices
- Multiplying matrices
- 2 × 2 Matrices and linear transformations
- Determinants of 2 × 2 matrices
- Inverses of 2 × 2 matrices
- Invariant points and lines in 2 dimensions
- 3 × 3 Matrices and linear transformations
- Determinants of 3 × 3 matrices
**Inverses of 3 × 3 matrices**- Matrices and simultaneous equations

# 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 **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.