All courses › Linear Algebra and Differential Equations › Inverse matrix

Inverse matrix

The inverse matrix $A^{-1}$ satisfies $AA^{-1}=I$ , and exists only when $\det A\neq0$ . It works much like "dividing by a matrix": it is used to solve $A\vec x=\vec b$ directly as $\vec x=A^{-1}\vec b$ . For a 2×2 matrix there is a simple formula.

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

inverse of

A=\begin{pmatrix}a&b\\c&d\end{pmatrix}

Symbols

I

the identity matrix (1 on the diagonal, 0 elsewhere)

Example

$A=\begin{pmatrix}2&1\\3&4\end{pmatrix}$ , $\det A=5$ : $A^{-1}=\frac15\begin{pmatrix}4&-1\\-3&2\end{pmatrix}$ .

Swap the diagonal, flip the sign of the other two entries, and divide everything by the determinant.

Practise matrices for free →

← Matrix multiplication · Eigenvalues and eigenvectors →

Part of Linear Algebra and Differential Equations: Matrices.