# Determinant of a Matrix

A determinant is a value that is useful in the analysis and solutions of systems of linear equations.

The determinant of the $2×2$ square matrix

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

is the value of the expression $ad-bc$ .

This number is useful in finding the inverse of the matrix , and in deciding whether the related system of equations

$\begin{array}{l}ax+by=0\\ cx+dy=0\end{array}$

has a solution (it does unless the determinant is equal to $0$ ).

The determinant of the $3×3$ matrix

$\left[\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\ {a}_{2}& {b}_{2}& {c}_{2}\\ {a}_{3}& {b}_{3}& {c}_{3}\end{array}\right]$

is the value of the expression ${a}_{1}{b}_{2}{c}_{3}-{a}_{1}{b}_{3}{c}_{2}+{a}_{2}{b}_{3}{c}_{1}-{a}_{2}{b}_{3}{c}_{1}+{a}_{3}{b}_{1}{c}_{2}-{a}_{3}{b}_{2}{c}_{1}$ .

Determinants for larger matrices can get a little ugly. Luckily, they rarely ask you to calculate them in high school!