3x3 Determinant Calculator

Reviewed by the CalculatorHub team, edited by James Graham. Cofactor expansion. Last verified 19 June 2026.

The determinant of a 3 by 3 matrix tells you whether it is invertible and how it scales volume. Enter the nine entries and the calculator expands along the first row, showing the three minor terms and the final determinant. A result of zero means the matrix is singular.

Row 1, col 1 (a)

Row 1, col 2 (b)

Row 1, col 3 (c)

Row 2, col 1 (d)

Row 2, col 2 (e)

Row 2, col 3 (f)

Row 3, col 1 (g)

Row 3, col 2 (h)

Row 3, col 3 (i)

Determinant

0.00

Term a(ei - fh)

0.00

Term -b(di - fg)

0.00

Term c(dh - eg)

0.00

Determinant formula

| a b c |
| d e f |
| g h i |
det = a(ei - fh) - b(di - fg) + c(dh - eg)

This is cofactor expansion along the first row. Each parenthesised term is a 2 by 2 minor determinant. A determinant of zero means the matrix is singular and has no inverse.

Worked example

For rows (1, 2, 3), (4, 5, 6), (7, 8, 10): det = 1 times (5 times 10 minus 6 times 8) minus 2 times (4 times 10 minus 6 times 7) plus 3 times (4 times 8 minus 5 times 7) = 1 times 2 minus 2 times minus 2 plus 3 times minus 3 = 2 plus 4 minus 9 = minus 3.

3x3 determinant: frequently asked questions

How do you find the determinant of a 3 by 3 matrix?

Expand along the first row: take a times the determinant of the bottom-right 2 by 2 minor, subtract b times its minor, add c times its minor. Each 2 by 2 determinant is the product of the main diagonal minus the product of the off-diagonal.

What is the full formula?

For rows (a, b, c), (d, e, f), (g, h, i): the determinant equals a(ei minus fh) minus b(di minus fg) plus c(dh minus eg). This is the standard cofactor expansion along the first row.

What does the determinant tell me?

It gives the signed volume scaling of the linear map the matrix represents. A determinant of zero means the matrix is singular and not invertible, with rows or columns that are linearly dependent. A nonzero determinant means the matrix is invertible.

Does the sign of the determinant matter?

Yes. A negative determinant means the transformation reverses orientation, like a reflection, while a positive one preserves it. The magnitude is the volume scale factor in either case.

Sources

NIST Digital Library of Mathematical Functions: Determinants and linear algebra.

Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.