Linear Algebra
Determinant
Type a square matrix into the live grid to get its determinant instantly — with an invertibility check, secondary stats, and every step of the cofactor expansion shown. Works for 2×2, 3×3, and 4×4 matrices.
Enter your matrix
3×3Matrix size
det(A) =-306
Invertible — non-singular
size
3×3
trace
11
invertible
yes
6 1 1; 4 -2 5; 2 8 7
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The basics
What is the determinant?
The determinant condenses an entire square matrix into one number. That number answers two big questions at once: can this matrix be inverted, and by how much does the transformation it describes stretch or shrink space? Everything else about the determinant follows from those two readings.
det(A) = 0 ⇒ no inverse
The invertibility test
A matrix has an inverse exactly when its determinant is non-zero. A zero determinant means the rows of A are linearly dependent — the transformation can't be undone.
det = 0 → singular; det ≠ 0 → invertible
|det(A)| = scaling factor
The geometric reading
The absolute value of the determinant is how much the matrix scales area in 2D or volume in 3D. A negative sign means the transformation also flips orientation, like a mirror.
|det| = 6 → areas scale ×6
Method
How to calculate it by hand
Small matrices have direct formulas; larger ones break down into smaller determinants.
- 1
The 2×2 formula
Multiply the main diagonal, multiply the anti-diagonal, then subtract. This is the base case every larger expansion bottoms out in.
det = ad − bc - 2
The 3×3 Rule of Sarrus
Add the three products running down-and-right, then subtract the three running down-and-left. Quick and reliable for 3×3 only.
(aei + bfg + cdh) − (ceg + afh + bdi) - 3
Cofactor expansion for n ≥ 3
Walk along the first row. Each entry a1j multiplies the determinant of the minor left after deleting its row and column, with alternating signs.
det(A) = Σ (−1)^(1+j) · a₁ⱼ · M₁ⱼ - 4
Recurse and sum
Each minor is itself a determinant, so keep expanding until you reach 2×2 base cases, then add the signed terms together.
det = T₁ + T₂ + … + Tₙ
Worked examples
See it at every size
The same idea scales from the two-line 2×2 formula up to a four-term cofactor expansion — switch tabs to follow each one through.
Given 3×3 matrix A
Rule of Sarrus — down-diagonals minus up-diagonals.
Down: (6)(−2)(7) + (1)(5)(2) + (1)(4)(8) = −42
Up: (1)(−2)(2) + (6)(5)(8) + (1)(4)(7) = 264
Subtract: −42 − 264
det(A) = −306
Where it shows up
Why the determinant matters
One number that decides whether systems are solvable, how shapes scale, and where curves bend.
Ax = b
Solving linear systems
Cramer's rule expresses each unknown as a ratio of determinants, and a non-zero determinant guarantees the system has one unique solution.
|det(J)|
Calculus & change of variables
The Jacobian determinant rescales integrals when you change coordinates — converting to polar or spherical form, for instance.
area = ½|det|
Geometry & graphics
The determinant of edge vectors gives the area of a parallelogram or the volume of a parallelepiped, and it tests whether points are collinear.
Properties
Rules worth remembering
A handful of identities make determinants far quicker to compute and reason about.
det(Aᵀ) = det(A)
Transpose
A matrix and its transpose share the same determinant, which is why you can expand along a row or a column and get the same answer.
det(AB) = det(A)·det(B)
Product rule
The determinant of a product is the product of the determinants — multiplying transformations multiplies their scaling factors. Work the product itself out first with the matrix calculator, then take the determinant of the result to check the identity.
det(A⁻¹) = 1 / det(A)
Inverse
The inverse undoes the scaling, so its determinant is the reciprocal. This only exists when det(A) is non-zero.
swap rows ⇒ × (−1)
Row swaps flip the sign
Interchanging two rows negates the determinant. Swap an even number of times and the sign returns to where it started.
triangular ⇒ ∏ diagonal
Triangular matrices
For an upper- or lower-triangular matrix the determinant is just the product of the diagonal entries — no expansion needed.
duplicate row ⇒ det = 0
Dependent rows
If two rows are equal, or one is a multiple of another, the matrix is singular and its determinant is zero.
Quick chart
2×2 determinant lookup
Set the top row, then read the determinant of every common 2×2 off the grid. Highlighted cells are singular — their determinant is zero, so the matrix has no inverse.
Top row of the 2×2:det = ad − bc
| c \ d | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| −2 | 2 | 3 | 4 | 5 | 6 | 7 |
| −1 | 0 | 1 | 2 | 3 | 4 | 5 |
| 0 | −2 | −1 | 0 | 1 | 2 | 3 |
| 1 | −4 | −3 | −2 | −1 | 0 | 1 |
| 2 | −6 | −5 | −4 | −3 | −2 | −1 |
| 3 | −8 | −7 | −6 | −5 | −4 | −3 |
Each cell is the determinant of [[1 2]; [c d]] for that row c and column d. Highlighted cells are singular (det = 0, no inverse).
Pitfalls
Common mistakes to avoid
A few slips trip up almost everyone learning determinants — watch for these.
Sarrus is 3×3 only
Don't stretch the Rule of Sarrus
The diagonal trick works for 3×3 matrices and nothing larger. For 4×4 and up you must use cofactor expansion or row reduction.
(−1)^(i+j)
Track the alternating signs
Cofactor signs follow a checkerboard, not a single rule. Forgetting to flip the sign on the second term is the most frequent expansion error.
det(A + B) ≠ det(A) + det(B)
The determinant isn't additive
Determinants multiply across products but do not split across sums. There is no shortcut for the determinant of a sum.
square only
Check the shape first
A determinant exists only for square matrices. If the row and column counts differ, there is nothing to compute.
FAQ
Common questions
The determinant is a single number computed from a square matrix. It summarizes how the linear transformation the matrix represents scales area or volume, and whether that transformation can be reversed. It is written det(A) or |A|.
For a 2×2 matrix [[a, b], [c, d]] the determinant is ad − bc: multiply the main diagonal, multiply the anti-diagonal, and subtract. For [[4, 7], [2, 6]] that is 4·6 − 7·2 = 24 − 14 = 10.
Use cofactor expansion along the first row, or the Rule of Sarrus: add the three down-diagonal products and subtract the three up-diagonal products. Both methods give the same result and the calculator shows every term.
A zero determinant means the matrix is singular — it has no inverse. Its rows (or columns) are linearly dependent, so the transformation collapses space to a lower dimension and cannot be undone.
Its absolute value is the factor by which area (2D) or volume (3D) is scaled. A non-zero value means the matrix is invertible; the sign tells you whether the transformation preserves or flips orientation.
No. The determinant is only defined for square matrices — those with the same number of rows and columns. A 2×3 or 3×4 matrix has no determinant.
Expand along a row or column into smaller minors (cofactor expansion), reducing a 4×4 to four 3×3 determinants, and so on recursively. Picking a row or column with zeros cuts the work. This calculator handles up to 4×4 directly.
A minor M₁ⱼ is the determinant of the submatrix left after deleting one row and column. The cofactor is that minor with a sign attached, (−1)^(i+j), which alternates across the checkerboard of positions.
Keep going
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Further reading
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Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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