Question

How do you simplify determinants $?$

Answer 1

To simplify determinant:

Simplification of determinant of a square matrix results in a scalar value.

If $A$ is a matrix, then determinant of $A$is denoted by $\left|A\right|$ or $detA$.

Suppose that $A$ is a $2\times 2$ matrix. Then the determinant of $A$is given by:

$\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|$

$\mathit{D}\mathit{e}\mathit{t}\mathbf{}\mathit{A}\mathbf{=}{\mathit{a}}_{\mathbf{11}}\mathbf{\times }{\mathit{a}}_{\mathbf{22}}\mathbf{-}{\mathit{a}}_{\mathbf{12}}\mathbf{\times }{\mathit{a}}_{\mathbf{21}}\mathbf{}$

where, ${\mathit{a}}_{\mathbf{11}}\mathbf{,}{\mathit{a}}_{\mathbf{12}}\mathbf{,}{\mathit{a}}_{\mathbf{21}\mathbf{,}}{\mathit{a}}_{\mathbf{22}\mathbf{}}$are elements of matrix $A$.

Example:

Take $A={\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]}_{2\times 2}$

$$\begin{gathered} detA=\left(1\right)\times \left(4\right)-\left(2\right)\times \left(3\right) \\ =4-6 \end{gathered}$$

$\Rightarrow$$\mathit{d}\mathit{e}\mathit{t}\mathbf{}\mathit{A}\mathbf{=}\mathbf{-}\mathbf{2}$