Matrix Determinant
Compute the determinant of any square matrix from 2×2 up to 6×6. Integer matrices are solved exactly; every result comes with the working shown and, for 2×2 and 3×3, a geometric picture.
What is a Determinant?
The determinant det(A) is a single number that tells you how the linear map A scales area or volume. A 2×2 matrix maps the unit square to a parallelogram whose area is |det A|; a 3×3 matrix maps the unit cube to a parallelepiped of volume |det A|.
The sign tells you about orientation: a negative determinant means the map flips space over, like a mirror. A determinant of zero means space is squashed into a lower dimension — the matrix has no inverse.
Geometric View
areaDeterminant Properties
| Operation | Effect on det |
|---|
Calculation Methods
Add the three “downward” diagonal products and subtract the three “upward” ones. Works only for 3×3 matrices.
Expand along any row or column; Mij is A with row i and column j removed. Pick the row with the most zeros to save work.
Reduce A to upper-triangular form U with row operations; s is the number of row swaps. This is how computers do it — fast even for large matrices.
Sources & Verification
- G. Strang, Introduction to Linear Algebra, 5th ed., Wellesley-Cambridge Press, 2016 — Ch. 5 (Determinants).
- D. C. Lay, S. R. Lay & J. J. McDonald, Linear Algebra and Its Applications, 5th ed., Pearson, 2016 — Ch. 3 (Determinants).