# Definition:Determinant/Matrix/Order 2

## Definition

Let $\mathbf A = \sqbrk a_2$ be a square matrix of order $2$.

That is, let:

$\mathbf A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$

Then the determinant of $\mathbf A$ is defined as:

$\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = \map \sgn {1, 2} a_{1 1} a_{2 2} + \map \sgn {2, 1} a_{1 2} a_{2 1} = a_{1 1} a_{2 2} - a_{1 2} a_{2 1}$

where $\sgn$ denotes the sign of the permutation.