How this calculator works
The determinant is a signed volume scale factor for the linear map represented by a square matrix. A non-zero determinant means the matrix is invertible. This calculator uses fraction-free elimination for the numeric result, then shows a cofactor expansion along the first row as a check. Determinants of larger matrices are computed from the same rules by recursion.
Formula
For 2 by 2, det = ad - bc. For 3 by 3, det = a(ei - fh) - b(di - fg) + c(dh - eg). In general, det(A) = sum over j of (-1)^(1+j) a1j det(M1j).
Worked example
The matrix with rows 2, 3 and 1, 4 has determinant 2x4 - 3x1 = 8 - 3 = 5. The matrix with rows 6, 1, 1; 4, -2, 5; 2, 8, 7 has determinant -306.
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Frequently asked questions
What does a zero determinant mean?
The matrix is singular: its rows or columns are linearly dependent, and no inverse exists. A linear system with that coefficient matrix may have no solution or infinitely many.
How does the determinant respond to row operations?
Adding a multiple of one row to another leaves it unchanged, swapping two rows negates it, and multiplying a row by k multiplies it by k.
Does the calculator show the working?
Yes. It shows a cofactor expansion along the first row beside the numeric determinant, so you can check the result by hand.