Find the determinant of a 2×2 or 3×3 matrix, and whether it is invertible. Enter the matrix with rows separated by semicolons.
Usage Tip
If any row or column is all zeros, or two rows are identical, the determinant is zero.
3×3: cofactor expansion across the top row
It is zero exactly when the matrix is singular (has no inverse), and otherwise gives the factor by which area or volume is scaled.
A non-zero determinant means the matrix is invertible.
A determinant of zero means the rows are linearly dependent.
The determinant is a single number computed from a square matrix that tells you whether the matrix can be inverted. A determinant of zero means the matrix is singular and has no inverse, while any other value means the inverse exists. For a 2 by 2 matrix it equals a times d minus b times c, and for a 3 by 3 matrix it is found by expanding along a row or column.
Geometrically, the absolute value of the determinant is the factor by which the matrix scales area in two dimensions or volume in three, and the sign tells you whether orientation is flipped. Swapping two rows changes the sign, and a matrix with two identical rows always has a determinant of zero.