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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

[ a b c d ]

is the value of the expression a d b c .

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

a x + b y = 0 c x + d y = 0

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

The determinant of the 3 × 3 matrix

[ a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 ]

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!