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

Arbitrary $3 \times 3$ Matrix
Let $\mathbf A$ be the matrix defined as:


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

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

Expanding row $2$:

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