3×3 Matrix Determinant Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The determinant of a 3×3 matrix is calculated by the rule of Sarrus: det = a(ei−fh) − b(di−fg) + c(dh−eg). For the matrix [[1,2,3],[0,1,4],[5,6,0]], the determinant is 1.
Explanation
The determinant of a square matrix is a single number that summarizes several essential geometric and algebraic properties of that matrix: it indicates in particular whether the associated system of linear equations has a unique solution (non-zero determinant) or not (zero determinant, as in our 2×2 matrix determinant calculator for the simpler two-dimensional case), and its absolute value corresponds to the volume of the parallelepiped formed by the three row (or column) vectors of the matrix. For a 3×3 matrix, the most direct method is expansion along the first row, often called the rule of Sarrus in its 3×3 special case: each coefficient of the first row is multiplied by the 2×2 determinant obtained by deleting its row and column, with alternating signs (+, −, +). A zero determinant signals a singular matrix, that is, non-invertible: its three rows (or columns) are linearly dependent, as the third test case of this calculator shows, where each row is offset by the same arithmetic progression from the previous one — the three vectors are then coplanar, hence a volume, and therefore a determinant, of zero. This notion of determinant is at the heart of Cramer's rule for solving a system of linear equations (see our system of linear equations with 2 unknowns calculator for the 2×2 case, where the principle is identical but simpler), and is also used to calculate the inverse of a matrix, which exists only if the determinant is non-zero.
Example: matrix [[1,2,3],[0,1,4],[5,6,0]]
Inputs
a=1, b=2, c=3, d=0, e=1, f=4, g=5, h=6, i=0.
Calculation
det = 1×(1×0 − 4×6) − 2×(0×0 − 4×5) + 3×(0×6 − 1×5) = 1×(−24) − 2×(−20) + 3×(−5) = −24 + 40 − 15 = 1.
Result
The determinant of this matrix is 1: it is invertible, and the associated linear system has a unique solution.
Frequently asked questions
What does a zero determinant mean concretely?
A zero determinant means the matrix is not invertible: its rows (or columns) are linearly dependent, that is, one can be expressed as a combination of the others. Geometrically, the three vectors of the matrix are then coplanar (they do not "fill" a three-dimensional space), and the associated system of linear equations has either no solution or infinitely many — never a unique solution.
Why do the signs alternate (+, −, +) in the formula?
This alternation of signs is part of the definition of expanding a determinant along a row or column (the cofactor of each term). It ensures the formula stays consistent with the fundamental properties of the determinant, notably that swapping two rows of a matrix changes the sign of its determinant — a property that would not hold with all-positive signs.
Does this formula work for larger matrices (4×4 and beyond)?
The cofactor-expansion principle generalizes to any size of square matrix, but its computational cost grows very quickly (factorially) with size — which makes it impractical by hand beyond 3×3 or 4×4. Computing software then uses more efficient methods, such as Gaussian elimination, to calculate the determinant of large matrices.