Knowing how to find a 3×3 determinant by hand makes the answer meaningful; checking it on a determinant calculator makes it trustworthy. We'll do both on one matrix and get the same number two different ways.
Our worked matrix throughout:
A = 1 2 3
0 1 4
5 6 0On this page
Method 1: Cofactor expansion (along row 1)
Take each entry of the top row, multiply it by the 2×2 determinant left when you delete that entry's row and column, and alternate the signs + − +:
det(A) = 1·|1 4; 6 0| − 2·|0 4; 5 0| + 3·|0 1; 5 6|Evaluate each 2×2 (top-left·bottom-right − top-right·bottom-left):
- |1 4; 6 0| = (1·0) − (4·6) = −24
- |0 4; 5 0| = (0·0) − (4·5) = −20
- |0 1; 5 6| = (0·6) − (1·5) = −5
So det(A) = 1·(−24) − 2·(−20) + 3·(−5) = −24 + 40 − 15 = 1.
Method 2: Rule of Sarrus
Copy the first two columns to the right, sum the three "down" diagonals, then subtract the three "up" diagonals:
1 2 3 | 1 2
0 1 4 | 0 1
5 6 0 | 5 6- Down: (1·1·0) + (2·4·5) + (3·0·6) = 0 + 40 + 0 = 40
- Up: (5·1·3) + (6·4·1) + (0·0·2) = 15 + 24 + 0 = 39
det(A) = 40 − 39 = 1. Same answer. ✔
Sarrus is a 3×3-only shortcut. For 4×4 and larger, fall back to cofactor expansion or row reduction.
Checking on the calculator
Switch to MATRIX mode and enter the rows:
1 2 3
0 1 4
5 6 0The calculator immediately reports det(A) = 1 — confirming both hand methods at once. It also shows the transpose and inverse, since a non-zero determinant guarantees the inverse exists.
What the answer means
A determinant of 1 here tells us A is invertible and preserves volume. Had it been 0, the matrix would be singular — no inverse, and dependent rows. That link between determinant and invertibility is the foundation of solving simultaneous equations with matrices. For the bigger picture, see the online matrix calculator pillar guide.
Frequently asked questions
What's the easiest way to find a 3×3 determinant?
Cofactor expansion along the first row is the most general; the rule of Sarrus is a faster 3×3-only shortcut.
Does Sarrus work for 4×4?
No — it's specific to 3×3. Use cofactor expansion or row reduction for larger matrices.
What does a determinant of zero mean?
The matrix is singular: dependent rows, no inverse, and no unique solution to its linear system.
Check your determinant
Enter your 3×3 matrix in MATRIX mode and read det(A) instantly.
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