The inverse and the determinant

For M = [[a, b], [c, d]], the determinant is $\det M = ad - bc$: the product of the main diagonal minus the product of the other diagonal. For [[5, 2], [3, 4]]: 5·4 − 2·3 = 14. The subtraction is the whole point; adding the diagonals, or multiplying all four entries, computes nothing meaningful.

Mind the signs when entries are negative: for [[3, −2], [1, 4]], the determinant is 3·4 − (−2)·1 = 12 + 2 = 14, not 12 − 2 = 10. The formula subtracts the product bc, whatever its sign turns out to be.

A matrix A is invertible when some matrix A⁻¹ satisfies $AA^{-1} = A^{-1}A = I$, the identity. The test is a one-liner: A is invertible exactly when det A ≠ 0.

The geometry says why. A 2 × 2 matrix transforms the plane, and |det| is its area scale factor. When det = 0 the matrix crushes the whole plane onto a line (area scale zero), and a crushed plane cannot be uncrushed: infinitely many points landed on top of each other, so no transformation can send them back. [[4, 2], [2, 1]] has det = 4 − 4 = 0; its columns point the same direction, everything lands on one line, and no inverse exists. A zero entry, by contrast, is harmless: [[2, 0], [0, 3]] has det = 6 and inverts fine.

When det = ad − bc ≠ 0, the inverse of [[a, b], [c, d]] is

$$A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

in words: swap the diagonal entries, negate the off-diagonal entries, divide everything by the determinant. For [[3, 1], [5, 2]], det = 6 − 5 = 1, so the inverse is [[2, −1], [−5, 3]]. Check: [[3, 1], [5, 2]] · [[2, −1], [−5, 3]] = [[6−5, −3+3], [10−10, −5+6]] = [[1, 0], [0, 1]]. That multiply-and-get-I check is always available and worth the thirty seconds.

The imposter is the entrywise reciprocal: replacing each entry with 1 over itself. Run the check on it and it fails immediately; for [[2, 4], [1, 3]], the reciprocal array [[1/2, 1/4], [1, 1/3]] multiplies in to [[5, 11/6], [7/2, 5/4]], nowhere near I. The true inverse is (1/2)[[3, −4], [−1, 2]].

As a transformation, M scales every region's area by |det M|. Under [[4, 1], [2, 3]] (det = 10), a region of area 2 lands with area 20. Determinant 1 preserves area; determinant 0 flattens it away.

The sign carries separate information: a negative determinant means the transformation flips orientation, like a reflection turning the plane over. [[0, 1], [1, 0]] has det = −1: it reflects across the line y = x, reversing every counterclockwise loop to clockwise while keeping areas intact. Negative does not mean shrinking, and it certainly does not mean non-invertible; only zero means that.

The inverse also earns its keep solving systems: if AX = B with A invertible, then X = A⁻¹B, applying the inverse on the left of both sides. With A = [[3, 1], [5, 2]] and B = ⟨1, 1⟩: X = [[2, −1], [−5, 3]] · ⟨1, 1⟩ = ⟨1, −2⟩, and A⟨1, −2⟩ = ⟨1, 1⟩ confirms it.