Matrix Determinant Calculator
Any square size, by elimination rather than cofactors.
Work out Matrix Determinant. Any square size, by elimination rather than cofactors. Correct where the obvious implementation is not.
Rows separated by semicolons or new lines, values by spaces or commas. Must be square.
Determinant of a 3×3 matrix
-306
Non-zero, so the matrix is invertible
The determinant is the factor by which the matrix scales area in 2D or volume in 3D. Zero means it collapses space onto a line or plane, which is exactly why a zero determinant makes a matrix non-invertible — the collapse cannot be undone. This uses LU decomposition with partial pivoting rather than cofactor expansion. Cofactors are what textbooks show and cost O(n!): a 10×10 matrix would take millions of times longer than elimination, which is O(n³). Pivoting also keeps the arithmetic stable when a leading entry is near zero. A negative determinant means the transformation also flips orientation — a reflection as well as a scaling.
How the Matrix Determinant Calculator works
Paste a square matrix for its determinant, trace and whether it is invertible. Computed by LU decomposition with partial pivoting, which is O(n³) — cofactor expansion is what textbooks show and costs O(n!), making it unusable past about 8×8.
Also known as: determinant of a 3x3 matrix · 4x4 determinant calculator · is a matrix invertible · singular matrix calculator
Frequently asked questions
What does the determinant mean?
The factor by which the matrix scales area in 2D or volume in 3D. A determinant of 3 triples areas; a negative one also flips orientation, adding a reflection to the scaling.
Why does a zero determinant mean the matrix is not invertible?
Because it collapses space onto a line or a plane, destroying a dimension. Information lost that way cannot be recovered, so no inverse transformation exists.
How is a 3×3 determinant calculated?
By the rule of Sarrus or cofactor expansion by hand. Computationally, by row reduction: eliminate to triangular form and multiply the diagonal, tracking sign changes from row swaps.
Why not use cofactor expansion?
It costs O(n!). A 10×10 matrix needs about 3.6 million times more work than elimination, and a 20×20 would outlast the universe. Elimination is O(n³) and handles both in microseconds.
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