Exercise 189Linear algebra › Matrices and linear systemsHigh schoolAdded 31 Oct 2026

Inverse of a 2×2 matrix

#inverse#determinants

Statement

Show that $A=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ is invertible and compute $A^{-1}$ .

Hint

For $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ with determinant $\Delta=ad-bc\ne0$ , the inverse is $\frac1\Delta\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$ .

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Solution

$\det A=1\cdot4-2\cdot3=-2\ne0$ , so $A$ is invertible, and
$A^{-1}=\frac1{-2}\begin{pmatrix}4&-2\\-3&1\end{pmatrix}=\begin{pmatrix}-2&1\\\tfrac32&-\tfrac12\end{pmatrix}.$

Inverse of a 2×2 matrix

Statement

Solution

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