Determinant & Inverse

By Victor Arce

A 2×2 matrix transforms space. Two questions capture how: by how much does it stretch areas, and can the transformation be undone? Both are answered by one number — the determinant.

Step through it on the grid below, watching the unit square.

The determinant is an area-scale factor

Apply a matrix and the unit square (area 1) becomes a parallelogram. Its area is the determinant:

det [[a, b], [c, d]] = a·d − b·c

For M = [[2, 1], [1, 2]] that's 2·2 − 1·1 = 3, so M triples every area. The determinant is a single number that says how a transformation grows or shrinks regions (and, if negative, that it flips orientation).

The inverse undoes the transformation

If M stretched space, the inverse M⁻¹ stretches it back so that M⁻¹ M leaves everything where it started. For a 2×2 matrix:

M⁻¹ = (1 / det M) · [[d, −b], [−c, a]]

The 1/det out front is the giveaway: undoing an area scale of 3 means scaling by 1/3, so det(M⁻¹) = 1 / det(M). In the animation, applying M⁻¹ snaps the parallelogram back to the original square.

When the determinant is zero

Look at [[2, 1], [2, 1]]: its columns point the same way, so it flattens the whole plane onto a single line. The unit square collapses to a segment — area 0, so det = 0. Once space is squashed flat, different starting points land on top of each other and you can't tell them apart, so there's no way to undo it: a zero determinant means the matrix is singular (non-invertible). Nonzero determinant ⇔ invertible is the rule to remember.

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