Inverse Matrix

Definition

A square matrix $\mathbf{A}$ is invertible (non-singular) if there exists a matrix ${\mathbf{A}}^{-1}$ such that

$$\mathbf{A}{\mathbf{A}}^{-1}={\mathbf{A}}^{-1}\mathbf{A}=\mathbf{I}$$

Intuition

The inverse undoes the transformation encoded by $\mathbf{A}$. Geometrically, if $\mathbf{A}$ stretches and rotates space, ${\mathbf{A}}^{-1}$ applies the exact reverse — restoring the original vectors. A matrix is invertible when it neither collapses any dimension to zero (full rank) nor maps multiple inputs to the same output (injective map), i.e. $\det \mathbf{A}\ne 0$.

Formal Description

For a $2\times 2$ matrix, solving $\mathbf{A}{\mathbf{A}}^{-1}=\mathbf{I}$ yields:

$${\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)}^{-1}=\frac{1}{ad-bc}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$$

The scalar $ad-bc=\det \mathbf{A}$ is the determinant of $\mathbf{A}$: the product of the diagonals minus the product of the off-diagonals.

• $\mathbf{A}$ is invertible if and only if $\det \mathbf{A}\ne 0$.
• $(\mathbf{AB}{)}^{-1}={\mathbf{B}}^{-1}{\mathbf{A}}^{-1}$
• If $\mathbf{A}$ is invertible, so is ${\mathbf{A}}^{\top }$, and $({\mathbf{A}}^{\top }{)}^{-1}=({\mathbf{A}}^{-1}{)}^{\top }$.

Applications

• Solving linear systems $\mathbf{Ax}=\mathbf{b}$: if $\mathbf{A}$ is invertible, then $\mathbf{x}={\mathbf{A}}^{-1}\mathbf{b}$.
• LU decomposition factorises $\mathbf{A}$ to compute solutions without explicitly forming ${\mathbf{A}}^{-1}$.

Trade-offs

• The inverse does not exist when $\det \mathbf{A}=0$ (singular matrix); the linear system $\mathbf{Ax}=\mathbf{b}$ then has either no solution or infinitely many.
• Explicitly computing ${\mathbf{A}}^{-1}$ is numerically expensive ($O(n^3)$) and unstable for ill-conditioned matrices; in practice, solve $\mathbf{Ax}=\mathbf{b}$ directly using LU or QR decomposition instead.
• For non-square matrices, only the pseudoinverse ${\mathbf{A}}^{+}$ exists; it minimises $\Vert \mathbf{Ax}-\mathbf{b}\Vert$ in the least-squares sense.

Links

• Matrix Operations
• Special Matrices
• LU Decomposition