Inverse of a Square Matrix (via Adjoint)

When Does an Inverse Exist?

A square matrix $A$ is called non-singular if $|A|\ne0$, and singular if $|A|=0$. Only a non-singular square matrix has an inverse: the inverse of $A$, written $A^{-1}$, is the unique matrix satisfying

If $A$ is singular, no such matrix can exist -- this follows directly from Section 5's key relation, since setting $|A|=0$ there would force $A(\operatorname{adj}A)=O$ (the zero matrix), which can never equal $I$.

The Inverse Formula

Section 5 established $A(\operatorname{adj}A)=(\operatorname{adj}A)A=|A|I$. Whenever $|A|\ne0$, both sides of this equation can be divided by the scalar $|A|$:

which -- by the very definition of an inverse just given -- identifies the bracketed matrix as $A^{-1}$ itself:

This single formula is the working method for finding the inverse of any $2\times2$ or $3\times3$ matrix in this course: compute $|A|$ (and check it is non-zero), compute every cofactor $C_{ij}$, assemble and transpose them into $\operatorname{adj}A$, and finally divide every entry of $\operatorname{adj}A$ by the scalar $|A|$.

Uniqueness of the Inverse

A matrix can have at most one inverse. If $B$ and $C$ were both inverses of $A$, then $B=BI=B(AC)=(BA)C=IC=C$, using associativity of matrix multiplication and the defining property of each inverse in turn -- so $B$ and $C$ must in fact be the same matrix. This is why the notation $A^{-1}$ (a single, specific matrix) is meaningful.

Worked Illustration ($2\times2$)

For $A=\begin{pmatrix}2&-1\\-3&4\end{pmatrix}$: $|A|=2(4)-(-1)(-3)=8-3=5\ne0$, so $A$ is invertible. Using the $2\times2$ shortcut of Section 5, $\operatorname{adj}A=\begin{pmatrix}4&1\\3&2\end{pmatrix}$, so

$A^{-1}=\frac15\begin{pmatrix}4&1\\3&2\end{pmatrix}.$ …