# Laplace's Expansion Theorem/Examples/Arbitrary 3x3 Matrix

## Example of Use of Laplace's Expansion Theorem

Let $\mathbf A$ be the matrix defined as:

$\mathbf A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$

Then $\map \det {\mathbf A}$ can be calculated using Laplace's Expansion Theorem as follows.

Expanding row $2$:

 $\ds \map \det {\mathbf A}$ $=$ $\ds \paren {-1}^{2 + 1} \times 4 \begin {vmatrix} 2 & 3 \\ 8 & 9 \end {vmatrix}$ $\ds$  $\, \ds + \,$ $\ds \paren {-1}^{2 + 2} \times 5 \begin {vmatrix} 1 & 3 \\ 7 & 9 \end {vmatrix}$ $\ds$  $\, \ds + \,$ $\ds \paren {-1}^{2 + 3} \times 6 \begin {vmatrix} 1 & 2 \\ 7 & 8 \end {vmatrix}$ $\ds$ $=$ $\ds - 4 \paren {2 \times 9 - 3 \times 8}$ $\ds$  $\, \ds + \,$ $\ds 5 \paren {1 \times 9 - 3 \times 7}$ $\ds$  $\, \ds - \,$ $\ds 6 \paren {1 \times 8 - 2 \times 7}$ $\ds$ $=$ $\ds 4 \paren {24 - 18} + 5 \paren {9 - 21} + 6 \paren {14 - 8}$ $\ds$ $=$ $\ds 0$

This shows that $\mathbf A$ is non-invertible.