Matrix Calculator · Practical guide
Determinant of a 3x3 Matrix
Use the first-row cofactor expansion: a(ei-fh) - b(di-fg) + c(dh-eg).
At a glance
The Formula
For a 3x3 matrix with first row a, b, c, second row d, e, f, and third row g, h, i, the determinant is:
det = a(ei - fh) - b(di - fg) + c(dh - eg)
Notice the alternating plus-minus-plus signs and that each term pairs a first-row element with the 2x2 determinant of what remains after deleting its row and column.
Interactive tool
Matrix Calculator
Enter your own values, then compare the result with the examples below.
Matrix A
Matrix B
Matrix A
Matrix B
Result
Worked Example
Take the matrix [[1, 2, 3], [0, 1, 4], [5, 6, 0]]:
- First term: 1 x (1x0 - 4x6) = 1 x (-24) = -24
- Second term: -2 x (0x0 - 4x5) = -2 x (-20) = +40
- Third term: 3 x (0x6 - 1x5) = 3 x (-5) = -15
Total: -24 + 40 - 15 = det = 1. Enter these nine values into the calculator above and it confirms the same result instantly.
What the Determinant Tells You
A nonzero determinant — like 1 here — means the matrix is invertible and its rows are linearly independent. A determinant of zero means the matrix collapses space and has no inverse. Determinants also appear in Cramer's rule, cross products, and volume scaling in transformations.
Frequently Asked Questions
Can I expand along a different row or column?
Yes — any row or column gives the same determinant if you apply the correct checkerboard of signs. The first row is simply the most common choice.
What is the rule of Sarrus?
A shortcut for 3x3 determinants: copy the first two columns to the right, then sum the three down-diagonals and subtract the three up-diagonals. It gives the same value as cofactor expansion.
Does the determinant change if I swap two rows?
Yes — swapping any two rows flips the sign of the determinant. Swapping twice restores the original value.
Is the determinant of a product the product of determinants?
Yes: det(A x B) = det(A) x det(B). So if det(A) = 1 and det(B) = -2, their product matrix has determinant -2.