3x3 Matrix Determinant Calculator
To find the determinant of a 3x3 matrix, multiply each entry in the first row by the 2x2 determinant left after deleting that entry's row and column, alternate the signs +, -, +, then add the three results. For rows (2,3,1), (0,4,5), (1,2,3), the determinant equals 15.
The Cofactor Expansion Formula for a 3x3 Determinant
For a 3x3 matrix A with entries a₁₁ through a₃₃, expanding along the first row gives:
det(A) = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁)
Each term pairs one entry from the first row with the 2x2 determinant of whatever is left after you cross out that entry's row and column. That 2x2 determinant is called a minor. The three minors used here are:
- M₁₁ = a₂₂a₃₃ − a₂₃a₃₂ (cross out row 1 and column 1)
- M₁₂ = a₂₁a₃₃ − a₂₃a₃₁ (cross out row 1 and column 2)
- M₁₃ = a₂₁a₃₂ − a₂₂a₃₁ (cross out row 1 and column 3)
So det(A) = a₁₁·M₁₁ − a₁₂·M₁₂ + a₁₃·M₁₃ — the same three minors this calculator displays alongside the final determinant. The middle term is subtracted; that single minus sign is where most errors happen.
Worked Example, Step by Step
Take A with rows (2, 3, 1), (0, 4, 5), (1, 2, 3). Enter these nine numbers above and the calculator returns the same three minors and the same determinant computed here by hand.
- M₁₁ = (4×3) − (5×2) = 12 − 10 = 2
- M₁₂ = (0×3) − (5×1) = 0 − 5 = −5
- M₁₃ = (0×2) − (4×1) = 0 − 4 = −4
det(A) = 2×2 − 3×(−5) + 1×(−4) = 4 + 15 − 4 = 15
Notice the second term: subtracting a negative (−5) turns into adding 15. That sign flip is exactly the step students skip when they rush.
The Rule of Sarrus: A Shortcut, and Its Limit
Sarrus's rule skips minors entirely. Copy the first two columns again to the right of the matrix, then add the three diagonal products running top-left to bottom-right, and subtract the three running top-right to bottom-left.
For the same matrix: det(A) = (2×4×3 + 3×5×1 + 1×0×2) − (1×4×1 + 2×5×2 + 3×0×3) = (24 + 15 + 0) − (4 + 20 + 0) = 39 − 24 = 15
Same answer as cofactor expansion, in fewer visible steps. The catch: the diagonal pattern only exists for 2x2 and 3x3 matrices. A 4x4 matrix has no set of diagonals that hits each row and column exactly once, so there is no Sarrus shortcut for it — you fall back to cofactor expansion or row reduction.
| Method | Works for | What you compute | Main risk |
|---|---|---|---|
| Cofactor expansion | Any n×n matrix | Three 2x2 minors, signed and summed | Forgetting the minus sign on the middle term |
| Rule of Sarrus | 2x2 and 3x3 only | Six diagonal products, added and subtracted | Trying to extend it to 4x4 or larger |
What It Means When the Determinant Is 0
A determinant of 0 means the matrix is singular: it has no inverse. Geometrically, the three row vectors don't fill out 3-D space — they collapse onto a plane, a line, or a point, so the parallelepiped they would form has zero volume.
This happens whenever one row (or column) is a combination of the others. Take rows (1, 2, 3), (2, 4, 6), (1, 1, 1): the second row is exactly twice the first, so
det = (1×4×1 + 2×6×1 + 3×2×1) − (3×4×1 + 1×6×1 + 2×2×1) = (4 + 12 + 6) − (12 + 6 + 4) = 22 − 22 = 0
When you land on 0, that's not a dead end — it's the answer. It tells you the rows are linearly dependent, and that a system built on that matrix has either no solution or infinitely many, never exactly one.
Common Mistakes When Computing a 3x3 Determinant
- Dropping the minus sign on the middle (a₁₂) term — the pattern is +, −, +, not +, +, +.
- Reading off a minor with the wrong row or column — cross out the row and column of the entry you're expanding on, nothing else.
- Mis-copying digits when rewriting the matrix for Sarrus's extra two columns; check each of the six diagonal products against the original grid before adding.
- Applying Sarrus's diagonal trick to a 4x4 matrix — it only works for 2x2 and 3x3, never larger.
- Treating a result of 0 as a mistake instead of a meaningful answer — a zero determinant correctly flags a singular matrix.
Frequently Asked Questions
What does the determinant of a 3x3 matrix tell you?
It's a single number packed with information: whether the matrix has an inverse (nonzero) or not (zero), how the linear transformation it represents scales volume, and whether its row or column vectors are linearly independent. A negative determinant means the transformation also flips orientation, like a mirror image.
How do you find the determinant of a 3x3 matrix by hand?
Expand along the first row: multiply a11 by the 2x2 determinant left after deleting its row and column, subtract a12 times its own 2x2 determinant, then add a13 times its 2x2 determinant. That is det(A) = a11*M11 - a12*M12 + a13*M13, using the three minors shown by this calculator.
What is the Rule of Sarrus and when can you use it?
Sarrus's rule copies the first two columns to the right of the matrix, then sums the three top-left-to-bottom-right diagonal products and subtracts the three top-right-to-bottom-left products. It's a fast mental shortcut, but it only works for 2x2 and 3x3 matrices - there is no valid version for 4x4 or larger.
Why doesn't the Rule of Sarrus work for 4x4 matrices?
The diagonal trick relies on exactly three top-left-to-bottom-right and three top-right-to-bottom-left diagonals that each touch one entry per row and column - that only happens for 3x3. A 4x4 matrix has diagonals that skip entries or double up, so the pattern no longer produces a correct determinant. Use cofactor expansion or row reduction instead.
What does it mean if the determinant is 0?
The matrix is singular - it has no inverse. Its rows (and columns) are linearly dependent, meaning at least one is a combination of the others, so the vectors collapse into a lower-dimensional space with zero volume. A linear system built on that matrix has either no solution or infinitely many, never exactly one.
Can you expand along a row or column other than the first row?
Yes. Cofactor expansion works along any row or column and gives the same determinant. Pick the one with the most zeros to cut down arithmetic - a zero entry makes that term vanish. Just keep the alternating sign pattern tied to position, not to the row you happened to choose.
What's the difference between a minor and a cofactor?
A minor M(i,j) is the determinant of the smaller matrix left after deleting row i and column j. A cofactor C(i,j) is that same minor multiplied by (-1)^(i+j) to apply the correct sign. In first-row expansion that is why the middle term (i=1, j=2) is subtracted: (-1)^(1+2) = -1.
Does swapping two rows change the determinant?
Yes - swapping any two rows (or columns) flips the sign of the determinant without changing its size. This matters if you rearrange a matrix to simplify a calculation: swap once and remember to multiply your final answer by -1, or you will get the right magnitude with the wrong sign.