Determinant Visualizer

2D — Oriented Area

For a 2×2 matrix \(A = [\mathbf{v}_1 \mid \mathbf{v}_2]\), the determinant equals the signed area of the parallelogram spanned by the two column vectors \(\mathbf{v}_1\) and \(\mathbf{v}_2\).

• Positive det: \(\mathbf{v}_2\) is counterclockwise from \(\mathbf{v}_1\) (right-hand orientation)
• Negative det: \(\mathbf{v}_2\) is clockwise from \(\mathbf{v}_1\) (orientation is reversed)
• Zero det: columns are collinear — parallelogram collapses to a line segment

3D — Oriented Volume

For a 3×3 matrix \(A = [\mathbf{v}_1 \mid \mathbf{v}_2 \mid \mathbf{v}_3]\), the determinant equals the signed volume of the parallelepiped spanned by the three column vectors.

• Positive det: right-handed orientation (like \(\hat{x}, \hat{y}, \hat{z}\))
• Negative det: left-handed (one vector reflected)
• Zero det: columns are linearly dependent — parallelepiped collapses

Key Formula

$$\det A = \text{signed area (2D) or signed volume (3D) of the parallelo-gram/piped}$$

Scale the matrix by \(k\): the area/volume scales by \(k^n\), so \(\det(kA) = k^n \det(A)\) for an \(n\times n\) matrix. Swapping two columns flips the sign.