Determinant Calculator

Math Linear Algebra • Determinants and Rank

Written by STEM Calculators Team Published February 22, 2026 Updated February 24, 2026

Compute the determinant \(\det(A)\) of an \(n\times n\) matrix using a small-matrix formula, cofactor expansion, or row operations (elimination). The sign flips on every row swap; \(\det(A)=0\) flags a singular matrix.

Size (n×n)

Method

Display decimals

Zero tolerance

Steps

Numbers like -3.5, 2e-4, or fractions like 7/3 are accepted. The canvas shows a signed-area parallelogram for \(2\times 2\) and elimination snapshots for larger matrices. Drag to pan • wheel to zoom • double-click to reset view.

Matrix \(A\)

2×2

Determinant

—

Ready

Visualization

Signed area / elimination snapshots.

matrix snapshot pivot cell pivot row

Enter a matrix and click “Calculate”.

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Theory

What a determinant tells you

The determinant \(\det(A)\) is a single number attached to a square matrix \(A\in\mathbb{R}^{n\times n}\). It summarizes three core ideas:

Invertibility: \(A\) is invertible exactly when \(\det(A)\neq 0\). If \(\det(A)=0\), the transformation collapses space into a lower dimension (singular).
Scaling (signed “volume”): \(|\det(A)|\) is the factor by which \(A\) scales area in 2D, volume in 3D, and hypervolume in higher dimensions.
Orientation: the sign of \(\det(A)\) indicates whether orientation is preserved (\(\det>0\)) or flipped (\(\det<0\)).

In \(2\times 2\), the columns of \(A\) form a parallelogram: its signed area equals \(\det(A)\). That is why the calculator shows a “volume sign” visual for 2D.

Direct formulas for \(2\times2\) and \(3\times3\)

For \[ A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}, \qquad \det(A)=ad-bc. \]

For \(3\times 3\), one standard method is cofactor expansion along the first row: \[ \det(A)=a_{11}(a_{22}a_{33}-a_{23}a_{32}) -a_{12}(a_{21}a_{33}-a_{23}a_{31}) +a_{13}(a_{21}a_{32}-a_{22}a_{31}). \]

These are fast and transparent for small matrices, but they do not scale well to large \(n\).

Cofactor expansion (expansion tree idea)

Cofactor expansion expresses \(\det(A)\) as a sum of smaller determinants (minors). Expanding along row \(i\) gives \[ \det(A)=\sum_{j=1}^{n} (-1)^{i+j} a_{ij}\,\det(M_{ij}), \] where \(M_{ij}\) is the \((n-1)\times(n-1)\) matrix obtained by removing row \(i\) and column \(j\).

Repeating this creates an expansion tree. It is great for understanding the definition, but it becomes computationally expensive as \(n\) grows. That’s why row operations are preferred for larger matrices.

Row operations method (efficient for larger \(n\))

For larger matrices, the practical approach is elimination: reduce \(A\) to an upper-triangular matrix \(U\). A key fact is: \[ \det(U)=u_{11}u_{22}\cdots u_{nn}. \]

The determinant reacts predictably to row operations:

Swap two rows: multiplies \(\det\) by \(-1\).
Add a multiple of one row to another: does not change \(\det\).
Multiply a row by \(k\): multiplies \(\det\) by \(k\).

The calculator’s row-ops method uses swaps (when needed) and row replacements. After elimination, \(\det(A)\) equals the product of the diagonal pivots times the swap sign \((−1)^{\#\text{swaps}}\). If a pivot becomes essentially zero, \(\det(A)=0\) and the matrix is singular.

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