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
is the value of the expression ad – bc.
ax + by = 0
cx + dy = 0
has a solution (it does unless the determinant is equal to 0).
The determinant of the 3 × 3 matrix
is the value of the expression a1b2c3 – a1b3c2 + a2b3c1 – a2b1c3 + a3b1c2 – a3b2c1.
Determinants for larger matrices can get a little ugly. Luckily, they rarely ask you to calculate them in high school!