Matrix Determinant Calculator

Math Linear Algebra • Matrices and Systems of Equations

Written by STEM Calculators Team Published February 21, 2026 Updated February 24, 2026

Compute the determinant \(\det(A)\) of an \(n\times n\) matrix. For \(2\times 2\) and \(3\times 3\) we show a direct formula. For larger matrices we use elimination (partial pivoting) and multiply pivots (tracking row swaps).

Size (n×n)

Method

Display decimals

Pivot tolerance

Steps

Play animates elimination snapshots once and stops automatically (pivot cell and pivot row highlighted). Drag to pan • wheel to zoom • double-click to reset.

Matrix \(A\)

2×2

det

Value

Computed determinant

—

Ready

Elimination preview (upper-triangular pivots)

The canvas shows the elimination snapshots used to compute \(\det(A)\) by pivot product (tracking row swaps).

matrix A snapshot pivot cell pivot row eliminated below pivot

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 important ideas:

Invertibility: \(A\) is invertible exactly when \(\det(A)\neq 0\). If \(\det(A)=0\), then the linear transformation “flattens” space into a lower dimension and cannot be reversed.
Scaling: \(|\det(A)|\) 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 the transformation preserves orientation (positive) or flips it (negative).

Because the determinant connects directly to invertibility, it is commonly placed right before the matrix inverse topic. Once you understand when \(\det(A)\) is nonzero, it becomes clear why some matrices have an inverse and others do not.

Direct formulas for small matrices

For \(2\times 2\) matrices, \[ A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}\quad\Rightarrow\quad \det(A)=ad-bc. \] This formula is fast and helps build intuition: the diagonal product \(ad\) is offset by the “cross” product \(bc\).

For \(3\times 3\) matrices, one common approach is expansion along the first row: \[ \det(A)=a_{11}(a_{22}a_{33}-a_{23}a_{32})-a_{12}(a_{21}a_{33}-a_{23}a_{31})+a_{13}(a_{21}a_{32}-a_{22}a_{31}). \] While correct, this becomes inefficient for large \(n\) because the number of terms grows quickly.

Elimination method (used for larger n)

For bigger matrices, the practical way to compute determinants is to convert \(A\) into an upper-triangular matrix \(U\) using elimination. Upper-triangular means all entries below the diagonal are zero. The key fact is: \[ \det(U)=u_{11}u_{22}\cdots u_{nn}\quad(\text{product of diagonal entries}). \]

During elimination, we use row operations that have predictable effects on the determinant:

Swap two rows: flips the sign of the determinant.
Add a multiple of one row to another: does not change the determinant.

(Scaling a row multiplies the determinant by that scalar, but the calculator avoids scaling rows for the determinant calculation.) With partial pivoting, we choose a stable pivot by selecting the largest available absolute entry in the current column. If no suitable pivot exists (the pivot is essentially zero), then the determinant is zero (or numerically indistinguishable from zero), and the matrix is singular.

The animation in the calculator shows these elimination snapshots and highlights the pivot position used at each stage. This makes it easier to connect the formula “product of pivots” to the row-reduction process.

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