Adjoint and Inverse of a Matrix

Adjoint of a Matrix

The adjoint of a square matrix $A$, written $\operatorname{adj}(A)$, is the transpose of the matrix of cofactors of $A$. That is, first replace every element of $A$ by its cofactor (Section 4) to get the cofactor matrix, then transpose that matrix.

For a $2\times2$ matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, this reduces to a quick shortcut:

— swap the two diagonal elements and change the sign of the two off-diagonal elements.

Singular and Non-Singular Matrices

A square matrix $A$ is singular if $|A|=0$, and non-singular if $|A| \ne 0$. Only a non-singular matrix has an inverse — this mirrors how, in ordinary arithmetic, only a nonzero number has a reciprocal.

Inverse of a Matrix

For a non-singular square matrix $A$, the inverse $A^{-1}$ is the unique matrix satisfying $AA^{-1}=A^{-1}A=I$, and it is computed from the adjoint by

$A^{-1} = \frac{1}{|A|}\,\operatorname{adj}(A)$ …