What is a matrix determinant?

The determinant is a single number computed from a square matrix that encodes important properties of the matrix. It tells you whether the matrix is invertible (det ≠ 0 means invertible; det = 0 means singular), by what factor the matrix scales areas or volumes when used as a linear transformation, and whether a system of linear equations has a unique solution. Determinants are fundamental in linear algebra, computer graphics, and physics.

How do you calculate a 2×2 determinant?

For a 2×2 matrix [[a, b], [c, d]], the determinant is det = ad − bc. Multiply the main diagonal (top-left to bottom-right: a × d) and subtract the product of the anti-diagonal (top-right to bottom-left: b × c). Example: [[3, 8], [4, 6]] → det = (3)(6) − (8)(4) = 18 − 32 = −14.

How do you calculate a 3×3 determinant?

For a 3×3 matrix [[a,b,c],[d,e,f],[g,h,i]], use cofactor expansion along the first row: det = a(ei − fh) − b(di − fg) + c(dh − eg). Each term is a matrix entry multiplied by its 2×2 minor determinant, with alternating signs (+, −, +). This calculator shows every step of the expansion.

What does it mean if the determinant is 0?

A determinant of zero means the matrix is singular — it has no inverse. Geometrically, the transformation collapses space: a 2×2 matrix with det = 0 maps the plane onto a line (or point), and a 3×3 matrix with det = 0 maps 3D space onto a plane or line. In linear algebra, a system of equations Ax = b has a unique solution only when det(A) ≠ 0. If det(A) = 0, the system is either inconsistent (no solution) or dependent (infinitely many solutions).

How is the determinant used in real applications?

Determinants appear across mathematics and engineering: solving systems of linear equations via Cramer's rule, finding matrix inverses (A⁻¹ = adj(A)/det(A)), computing cross products in 3D geometry (used in physics and computer graphics), determining whether a set of vectors is linearly independent, and calculating the volume scaling factor of linear transformations. In computer graphics, the sign of the determinant indicates whether a transformation reverses orientation (flips the image).

What does a determinant of 0 mean practically?

A zero determinant means the matrix is singular and has no inverse. In practical terms: if you are solving a system of linear equations Ax = b, det(A) = 0 means the system either has no solution (the equations contradict each other) or infinitely many solutions (the equations are redundant). In 3D graphics, a transformation matrix with det = 0 flattens 3D space onto a plane or line, destroying information. In engineering, a stiffness matrix with det = 0 indicates a structural mechanism.

What is Cramer's rule?

Cramer's rule uses determinants to solve a linear system Ax = b. For each variable xᵢ, replace the i-th column of A with the vector b, compute the determinant of the modified matrix, and divide by det(A). For a 2×2 system: x = det([b₁, a₁₂; b₂, a₂₂]) / det(A) and y = det([a₁₁, b₁; a₂₁, b₂]) / det(A). Cramer's rule is theoretically elegant but computationally slow for large systems — Gaussian elimination is preferred in practice for systems larger than 3×3.

Determinant Calculator — 2×2 and 3×3 Matrix

Calculates the determinant of any 2×2 or 3×3 matrix with step-by-step cofactor expansion and an invertibility check.

How to Use the Determinant Calculator

This determinant calculator works for 2×2 and 3×3 matrices — select the size, then fill in all entries. The determinant is calculated instantly as you type. Below the result cards you will find a full step-by-step cofactor expansion showing every intermediate product. The invertibility card tells you at a glance whether the matrix has an inverse. Use the Share button to save a link with your matrix pre-filled.

If you need to go further and actually find the inverse of a matrix or reduce it to row echelon form, the process requires additional steps beyond the determinant alone. For cross products of 3D vectors — which use a determinant-like structure — see our cross product calculator.

The Determinant Formulas

The determinant formula depends on the matrix size. Both are shown below with worked examples.

2×2 Determinant Formula

For a 2×2 matrix with entries a, b (top row) and c, d (bottom row):

det([[a, b], [c, d]]) = ad − bc

Example: det([[5, 3], [2, 7]]) = (5)(7) − (3)(2) = 35 − 6 = 29. Since 29 ≠ 0, the matrix is invertible.

3×3 Determinant Formula (Cofactor Expansion)

For a 3×3 matrix [[a, b, c], [d, e, f], [g, h, i]], expand along the first row:

det = a(ei − fh) − b(di − fg) + c(dh − eg)

Each term in parentheses is the 2×2 determinant of the submatrix formed by deleting the first row and the column of the corresponding entry. The signs alternate: +, −, +.

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What the Determinant Tells You

The determinant encodes several critical properties of a matrix:

Invertibility — a matrix is invertible (has a unique inverse) if and only if its determinant is non-zero. If det = 0, the matrix is called singular and cannot be inverted. This directly affects whether a system of linear equations Ax = b has a unique solution.
Area and volume scaling — a 2×2 matrix with |det| = k scales areas by a factor of k. A 3×3 matrix scales volumes by |det|. For instance, if a 2×2 transformation has det = 3, it triples all areas in the plane.
Orientation — the sign of the determinant indicates whether the transformation preserves (positive det) or reverses (negative det) orientation. In 2D, a negative determinant means the transformation includes a reflection.
Linear independence — a set of n vectors is linearly independent if and only if the matrix formed by those vectors as columns has a non-zero determinant.

Step-by-Step 3×3 Example

Let the matrix be [[2, 1, 3], [0, −1, 4], [5, 2, −2]]:

Minor for a = 2: det([[−1, 4], [2, −2]]) = (−1)(−2) − (4)(2) = 2 − 8 = −6
Minor for b = 1: det([[0, 4], [5, −2]]) = (0)(−2) − (4)(5) = 0 − 20 = −20
Minor for c = 3: det([[0, −1], [5, 2]]) = (0)(2) − (−1)(5) = 0 + 5 = 5
det = 2(−6) − 1(−20) + 3(5) = −12 + 20 + 15 = 23

Since det = 23 ≠ 0, this matrix is invertible. For problems involving the hypotenuse or distances in geometry, our Pythagorean theorem calculator handles the geometric side.

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Determinant Properties

Several algebraic properties make determinants easier to compute by hand — and explain why linear algebra software uses row reduction rather than cofactor expansion for large matrices:

det(AB) = det(A) · det(B) — the determinant of a product equals the product of the determinants.
det(Aᵀ) = det(A) — the determinant is unchanged by transposition.
det(kA) = kⁿ · det(A) — scaling a matrix by k scales the determinant by kⁿ (where n is the matrix size).
Row swap flips the sign — swapping any two rows multiplies the determinant by −1.
Row with all zeros → det = 0 — if any row or column is entirely zero, the determinant is zero and the matrix is singular.
Identical rows → det = 0 — two identical rows (or columns) always give a determinant of zero.

For large matrices (4×4 and above), most software uses Gaussian elimination (row reduction) rather than cofactor expansion, because cofactor expansion requires O(n!) operations while row reduction runs in O(n³). For this reason the determinant calculator covers 2×2 and 3×3 matrices, which are the sizes most commonly needed for coursework and hand calculations. For solving the same systems with row reduction, our RREF calculator shows full step-by-step Gaussian elimination.

Sources & References

Matrix Determinants — Linear Algebra — Khan Academy
Determinant — Wolfram MathWorld — Wolfram Research

Determinant Calculator