Inverse of a 2x2 matrix

For square $2\times 2$ matrices there is a handy explicit formula for the inverse, which generalises the idea of the “reciprocal” of a number.

Property — Inverse of a $2\times 2$ matrix

Let $A = \begin{pmatrix} a & b \\ c & d\end{pmatrix}$ with $\det A = ad-bc\ne 0$. There exists a unique inverse matrix $A^{-1}$ with $AA^{-1}=A^{-1}A=I_2$, given by: $A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a\end{pmatrix}.$ If $\det A=0$, the matrix is not invertible (it corresponds to a system with proportional rows). The number $\det A$ is precisely the $\Delta$ of Cramer’s method.

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Topics: Linear systems
Concepts: Determinant · Inverse matrix
Methods: 2x2 inverse matrix
Skills: Use formulae