Matrix Inverse

The matrix that undoes a linear transformation — when it exists, how to find it, and what its non-existence means for a system.

A⁻¹ undoes A — applying both in sequence returns the original vector

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Definition

The inverse of a square matrix $A$ is the matrix $A^{-1}$ such that:

$AA^{-1} = A^{-1}A = I$

where $I$ is the identity matrix. $A^{-1}$ exists iff $A$ is invertible (also called nonsingular), which happens iff $\det(A) \neq 0$ .

2×2 formula: for $A = \begin{pmatrix}a&b\\c&d\end{pmatrix}$ with $\det(A) = ad-bc \neq 0$ :

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

Why it matters: $A^{-1}$ lets you solve $A\mathbf{x} = \mathbf{b}$ as $\mathbf{x} = A^{-1}\mathbf{b}$ .

Key properties

The inverse, if it exists, is unique — a matrix can't have two different inverses
$(A^{-1})^{-1} = A$ — inverting twice returns the original matrix
Only square matrices can have a two-sided inverse in this sense
$I^{-1} = I$ — the identity is its own inverse

Common mistakes

Assuming every square matrix is invertible: $\det(A) = 0$ means no inverse exists, no matter how the matrix looks otherwise
Computing $A^{-1}$ just to solve $A\mathbf{x}=\mathbf{b}$ : this is slower and less numerically accurate than directly solving the system (e.g. solve(A,b) instead of inv(A) @ b) — explicit inversion is rarely the right tool

Solving a system via the inverse

$A = \begin{pmatrix}2&1\\5&3\end{pmatrix}$ , solve $A\mathbf{x} = \begin{pmatrix}4\\11\end{pmatrix}$ .

$\det(A) = 6-5 = 1$ . $A^{-1} = \begin{pmatrix}3&-1\\-5&2\end{pmatrix}$ .

$\mathbf{x} = A^{-1}\mathbf{b} = \begin{pmatrix}3&-1\\-5&2\end{pmatrix}\begin{pmatrix}4\\11\end{pmatrix} = \begin{pmatrix}1\\2\end{pmatrix}$ .

Try it

Find $A^{-1}$ for $A = \begin{pmatrix}3&2\\1&1\end{pmatrix}$ and verify $AA^{-1} = I$ .

Solution

$\det(A) = 3-2 = 1$ . $A^{-1} = \begin{pmatrix}1&-2\\-1&3\end{pmatrix}$ .

Verify: $\begin{pmatrix}3&2\\1&1\end{pmatrix}\begin{pmatrix}1&-2\\-1&3\end{pmatrix} = \begin{pmatrix}3-2&-6+6\\1-1&-2+3\end{pmatrix} = \begin{pmatrix}1&0\\0&1\end{pmatrix}$ ✓

Related concepts

Needs first

Determinants Gaussian Elimination Matrices

Linear Transformations Matrix Multiplication Pseudoinverse Rank

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