Matrix Determinant Calculator

See the matrix. Follow the algebra. Understand what the determinant means.

Enter matrix A

Real entries

A =

a₁₁a₁₂a₂₁a₂₂

Enter rows from left to right. Negative values and decimals are supported.

Paste a matrix

Paste rows separated by line breaks, with entries separated by commas, spaces or tabs.

Result display

Precision & calculation notes

Finite decimal entries are interpreted exactly as rational numbers. Rank and singularity use exact arithmetic, not a floating-point tolerance. Maximum 60 significant digits per entry; scientific exponents −100 to 100.

Matrix position matters.
The first column is u = (1, 3); the second is v = (2, 4). These are the vectors shown in the geometry panel.

DETERMINANT · det(A)−2

Nonzero determinant · the inverse exists.

Exact determinant

Trace · tr(A)5

Matrix rank2 / 2

Matrix statusInvertible

Geometry of your determinant

2D transformation

Unit square · area 1Image under A · area 2

Shared x/y unit scale. The shaded parallelogram is the image of the unit square under x ↦ Ax.

Step-by-step determinant

2 × 2 · cross products

1Main diagonalad = 1 × 4 = 4

2Other diagonalbc = 2 × 3 = 6

3Subtractdet(A) = 4 − 6 = −2

Inverse matrix · A⁻¹

Exists

A⁻¹ =

−2132−12

VerificationA × A⁻¹ = I₂

For a 2 × 2 matrix, A⁻¹ = (1/det A) [[d, −b], [−c, a]]. An inverse is unavailable when det(A) = 0.

A look inside the 3 × 3 method

Independent example B · switch to 3 × 3 above to analyze your own matrix.

Sarrus rule

On narrow screens, swipe the diagram horizontally.

↘ Positive group↙ Subtracted group

Follow the six products

Downward(2 × 3 × 2) + (1 × 1 × 1) + (0 × 0 × 0)12 + 1 + 0 = 13

Upward(0 × 3 × 1) + (1 × 0 × 2) + (2 × 1 × 0)0 + 0 + 0 = 0

det(B) = 13 − 0 = 13

Cofactor check along row 1:
2(3 × 2 − 1 × 0) − 1(0 × 2 − 1 × 1) + 0 = 13.

Swap two rowsThe determinant changes sign.

Dependent rowsdet(A) = 0; the matrix is singular.

Triangular matrixMultiply the diagonal entries.

Formula & interpretation

2 × 2: det(A) = ad − bc

Trace: sum of the main-diagonal entries.
Rank: number of independent columns or rows.
Invertible: det(A) ≠ 0 for a square matrix.

|det(A)| scales area in 2D and volume in 3D. The sign records orientation; it is not a quality or stability score.

Frequently asked questions

Can a determinant be negative?

Yes. A negative determinant describes an orientation-reversing transformation. Geometric area and volume scale by its absolute value.

Does a small determinant prove a matrix is nearly singular?

Not by itself. The determinant depends strongly on scale. Conditioning and numerical rank require appropriate scale-aware analysis.

Is Sarrus valid for every matrix size?

No. The diagonal-copy rule is specific to 3 × 3 determinants. Cofactor expansion and elimination generalize to larger sizes.

What happens when the determinant is zero?

The square matrix is singular, its columns are dependent and a unique inverse does not exist.

Sources & Reference Guides

NISTDigital Library of Mathematical FunctionsDeterminants, minors, cofactors and matrix operators.MITOpenCourseWare · Linear AlgebraInverse matrices, determinants and volume.

Independent educational references · No institutional endorsement implied.

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