Matrix Inverse

Geometry and algebra of linear maps, vectors, and matrices

The inverse A⁻¹ is the transformation that undoes A. Apply A then A⁻¹ and every vector returns home: A⁻¹A = AA⁻¹ = I. If A rotates 30°, its inverse rotates back 30°; if A doubles lengths, its inverse halves them.

Not every matrix can be undone. An inverse exists only when A is full rank, equivalently when its determinant is nonzero. The reason is geometric: if A flattens space (collapsing a direction to zero, as a low-rank matrix does), information is destroyed and there is no way to reconstruct it. Such a matrix is singular.

For a 2×2 matrix there is a memorable closed form. Swap the diagonal, negate the off-diagonal, divide by the determinant:

Where this lives in MLThe inverse is conceptually central but practically avoided. The normal equations of regression are written β = (XᵀX)⁻¹Xᵀy, yet real solvers never form that inverse; they solve the system directly because inverting is costly and numerically fragile. Knowing when a matrix is invertible (full rank) tells you whether your problem is well-posed or degenerate.

▶ Matrix Inverse

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