Inverse & Determinant
Every matrix represents a transformation. But can it be reversed? The inverse and the determinant answer this in two complementary ways — one algebraically, the other geometrically.
Undoing a Transformation
The inverse A⁻¹ satisfies a two-sided identity:
Applying A then A⁻¹ — or A⁻¹ then A — brings you back to where you started. Only some matrices have an inverse; those that don't are called singular.
When Does It Exist?
These three conditions are equivalent — any one implies the other two. A singular matrix "collapses" space, making reversal impossible.
The Inverse Formula
For a 2×2 matrix, the inverse is compact:
Recipe: swap the diagonal (a↔d), negate off-diagonal (b, c), divide by det = ad − bc.
Row Reduction to Invert
For any size: form the augmented matrix [A | I] and row-reduce.
Rules of Inversion
- (AB)⁻¹ = B⁻¹A⁻¹ — order reverses (shoes and socks rule)
- (Aᵀ)⁻¹ = (A⁻¹)ᵀ — transpose and inverse commute
- (A⁻¹)⁻¹ = A — inverting twice recovers the original
- (cA)⁻¹ = (1/c)A⁻¹ — scalar factors invert
Determinant = Volume Scale
|det(A)| measures the factor by which the transformation scales areas (2D) or volumes (3D). The sign tells you whether orientation is preserved (positive) or flipped (negative). det = 0 means collapse — no inverse possible.
2×2 and 3×3 Formulas
For 3×3: use cofactor expansion along any row. Each entry multiplies a 2×2 minor with alternating ± signs. This extends to any dimension recursively.
Determinant Rules
- det(Aᵀ) = det(A)
- det(A⁻¹) = 1/det(A)
- Row swap negates det; row scale multiplies det
Key Takeaways
- A⁻¹ exists ↔ A is square, full rank, and det(A) ≠ 0
- 2×2 inverse: swap diagonal, negate off-diagonal, divide by det
- General inverse: row-reduce [A | I] to get [I | A⁻¹]
- det measures volume scaling; sign indicates orientation flip
- det(AB) = det(A)·det(B) — volume factors compose multiplicatively
Module 3: Systems of Equations
With A⁻¹ in hand, solving Ax = b is conceptually just x = A⁻¹b. But computing inverses is expensive. Module 3 shows how Gaussian elimination solves systems directly — faster, more stable, and revealing the structure of solutions when they're not unique.