Last updated: 2026-07-01

What this calculator does

This calculator finds the determinant of a square matrix — a single number that captures how the matrix stretches, squashes, or flips space. Choose 2×2 or 3×3 mode, enter the entries, and the determinant updates along with a step-by-step derivation.

The determinant of a matrix A is written $\det A$ or $|A|$.

The 2×2 determinant

For a 2×2 matrix, the determinant is the product of the main diagonal minus the product of the anti-diagonal:

$$\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$$

Worked example (2×2)

Take the matrix [[1, 2], [3, 4]]:

$\det A = (1)(4) - (2)(3) = 4 - 6 = -2$

The negative sign means this transformation reverses orientation, and the magnitude 2 means it doubles area.

The 3×3 determinant by cofactor expansion

For a 3×3 matrix, expand along the first row. Each first-row entry is multiplied by the determinant of the 2×2 minor that remains after deleting that entry's row and column, with alternating signs + − +:

$$\det \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} = a_{11} M_{11} - a_{12} M_{12} + a_{13} M_{13}$$

where the minors are

$$M_{11} = \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix}, \quad M_{12} = \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix}, \quad M_{13} = \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}.$$

Worked example (3×3)

Take the matrix [[6, 1, 1], [4, −2, 5], [2, 8, 7]]:

$M_{11} = (-2)(7) - (5)(8) = -14 - 40 = -54$

$M_{12} = (4)(7) - (5)(2) = 28 - 10 = 18$

$M_{13} = (4)(8) - (-2)(2) = 32 + 4 = 36$

Now combine with the first-row entries and alternating signs:

$\det A = (6)(-54) - (1)(18) + (1)(36) = -324 - 18 + 36 = -306$

The rule of Sarrus (3×3 shortcut)

An alternative for 3×3 matrices only is the rule of Sarrus: copy the first two columns to the right of the matrix, add the three full left-to-right diagonal products, and subtract the three right-to-left diagonal products:

$$\det A = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33}$$

This gives the same answer as cofactor expansion. Sarrus is quick to apply by hand, but it does not generalize beyond 3×3, which is why cofactor expansion is the method taught first.

When the determinant is zero

A determinant of zero signals a singular matrix:

• Its rows (and columns) are linearly dependent.
• The transformation collapses space onto a line or plane, losing a dimension.
• The matrix has no inverse.
• A linear system Ax = b with a singular A has either no solution or infinitely many.

For example, [[1, 2, 3], [4, 5, 6], [7, 8, 9]] has determinant 0 because each row is an arithmetic step from the one above, making the rows linearly dependent.

Geometric meaning

The determinant measures signed scaling:

• 2×2: the determinant equals the signed area of the parallelogram spanned by the two column vectors. A determinant of 5 means areas are scaled by 5; a determinant of −5 means areas are scaled by 5 and orientation is flipped.
• 3×3: the determinant equals the signed volume of the parallelepiped spanned by the three column vectors.
• A magnitude of 1 means area or volume is preserved (a rotation or reflection); a magnitude of 0 means the figure is flattened to zero area or volume.

Useful properties

• $\det(AB) = \det(A)\,\det(B)$ — the determinant of a product is the product of determinants.
• $\det(A^{\mathsf{T}}) = \det(A)$ — transposing leaves the determinant unchanged.
• Swapping two rows flips the sign of the determinant.
• Multiplying one row by a scalar $k$ multiplies the determinant by $k$.
• The determinant of a triangular matrix is the product of its diagonal entries.

Frequently Asked Questions (FAQ)

What is the determinant of a matrix?

The determinant is a single number computed from a square matrix that summarizes key properties of the linear transformation it represents. Its absolute value tells you how much the transformation scales area (2×2) or volume (3×3), and its sign tells you whether orientation is preserved or flipped. A determinant of zero means the matrix is singular and has no inverse.

What does a determinant of zero mean?

A zero determinant means the matrix is singular: its rows (or columns) are linearly dependent, so the transformation collapses space into a lower dimension. A singular matrix has no inverse, and a linear system with a singular coefficient matrix has either no solution or infinitely many.

What is the difference between cofactor expansion and the rule of Sarrus?

Both give the same 3×3 determinant. Cofactor expansion multiplies each first-row entry by the determinant of its 2×2 minor with alternating signs — it generalizes to any size. The rule of Sarrus is a shortcut that sums the three left-to-right diagonal products and subtracts the three right-to-left ones; it works only for 3×3 matrices. This calculator shows the cofactor method because it carries over to larger matrices.

What does the determinant mean geometrically?

For a 2×2 matrix, the determinant equals the signed area of the parallelogram spanned by its column vectors. For a 3×3 matrix, it equals the signed volume of the parallelepiped spanned by its three columns. A negative sign indicates the transformation reverses orientation (a reflection), and a magnitude of one means area or volume is preserved.