Determinant Calculator
Find the determinant of a 2x2 or 3x3 matrix with step-by-step work. The determinant tells you whether a matrix is invertible and plays a key role in solving systems of linear equations, computing eigenvalues, and more.
2x2 Determinant
For a 2x2 matrix [[a, b], [c, d]], the determinant is:
det = ad - bc
Example: [[3, 7], [1, 5]] has det = (3)(5) - (7)(1) = 15 - 7 = 8.
3x3 Determinant (Cofactor Expansion)
For a 3x3 matrix, expand along the first row:
det = a11(a22 a33 - a23 a32) - a12(a21 a33 - a23 a31) + a13(a21 a32 - a22 a31)
Each 2x2 block in parentheses is called a minor. The alternating +/- signs come from the cofactor sign pattern.
What the Determinant Tells You
- Non-zero determinant: The matrix is invertible (has a unique inverse). A system of equations with this coefficient matrix has exactly one solution.
- Zero determinant: The matrix is singular (not invertible). The corresponding system has either no solution or infinitely many solutions.
- The absolute value of the determinant also represents the scale factor by which the matrix transforms areas (2D) or volumes (3D).