How it works
The determinant is a single number computed from a square matrix. For a 2×2 matrix [a b; c d] it is ad − bc. For a 3×3 matrix it is found by cofactor expansion along the first row: each entry times the determinant of the 2×2 matrix left after deleting its row and column, with alternating signs.
A determinant of zero means the matrix is singular: it has no inverse, and a system of equations using it has no unique solution. The absolute value of the determinant is the factor by which the matrix scales area (2×2) or volume (3×3).
Formula
2×2: ad − bc. 3×3: a(ei − fh) − b(di − fg) + c(dh − eg)
Questions people ask
What does a zero determinant mean?
The rows (and columns) are linearly dependent, so the matrix squashes space into a lower dimension, has no inverse, and the related linear system has either no solution or infinitely many.
What is the rule of Sarrus?
A shortcut for 3×3 determinants only: add the products of the three down-right diagonals and subtract the three up-right diagonals. It gives the same answer as cofactor expansion.
What happens to the determinant if I swap two rows?
It changes sign. Multiplying a row by k multiplies the determinant by k, and adding a multiple of one row to another leaves it unchanged.