3x3 Matrix Determinant Calculator
Compute the determinant of any 3×3 matrix by cofactor expansion, with the three 2×2 minors shown so you can follow every step.
How it works
det = a(ei − fh) − b(di − fg) + c(dh − eg)
A 3×3 determinant is found by cofactor expansion along the top row: each entry is multiplied by the 2×2 determinant of the block left when you delete its row and column, with the middle term subtracted because of the alternating sign pattern. A non-zero determinant means the matrix is invertible and its rows are linearly independent; a determinant of zero means the matrix is singular.
Worked example
For rows (1, 2, 3), (0, 4, 5), (1, 0, 6): det = 1·(4·6 − 5·0) − 2·(0·6 − 5·1) + 3·(0·0 − 4·1) = 1·24 − 2·(−5) + 3·(−4) = 24 + 10 − 12 = 22.
Sources: Determinant (Wikipedia)
Frequently asked questions
What does a determinant of zero mean?
The matrix is singular: it has no inverse, its rows (and columns) are linearly dependent, and it collapses volume to zero. A linear system with a zero coefficient determinant has either no solution or infinitely many.
Does the sign of the determinant matter?
Yes. Its absolute value is the volume scaling factor of the linear map, and its sign tells you whether the map preserves orientation (positive) or flips it (negative).
Can I expand along a different row or column?
You can — cofactor expansion gives the same determinant along any row or column. Expanding along one containing zeros is the least work; this tool expands along the first row.
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This calculator is for educational purposes. Double-check important results before acting on them.