Definition:Determinant/Matrix/Order 2

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Definition

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

That is, let:

$\mathbf A = \begin {bmatrix} a_{1 1} & a_{1 2} \\ a_{2 1} & a_{2 2} \end {bmatrix}$


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

\(\displaystyle \begin {vmatrix} a_{1 1} & a_{1 2} \\ a_{2 1} & a_{2 2} \end{vmatrix}\) \(=\) \(\displaystyle \map \sgn {1, 2} a_{1 1} a_{2 2} + \map \sgn {2, 1} a_{1 2} a_{2 1}\)
\(\displaystyle \) \(=\) \(\displaystyle a_{1 1} a_{2 2} - a_{1 2} a_{2 1}\)


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


Sources