Adjoint of a Square Matrix

Definition

For a square matrix $A$, first form the matrix of cofactors of $A$ -- the matrix obtained by replacing every entry $a_{ij}$ with its cofactor $C_{ij}$ (Section 3). The adjoint of $A$, written $\operatorname{adj}A$, is defined as the transpose of this matrix of cofactors:

Notice carefully that the cofactors are transposed -- the cofactor $C_{ij}$ of the original entry $a_{ij}$ ends up in position $(j,i)$ of $\operatorname{adj}A$, not position $(i,j)$.

Shortcut for a $2\times2$ Matrix

For $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, computing all four cofactors and transposing gives the compact rule

swap the two diagonal entries $a$ and $d$, and change the sign of the two off-diagonal entries $b$ and $c$. This shortcut is worth memorising directly, since it avoids computing four separate $1\times1$ minors for what is, in the end, always this same simple pattern.

The Key Relation: $A(\operatorname{adj}A)=(\operatorname{adj}A)A=|A|\,I$

This is the single most important fact about the adjoint, and the reason it is introduced at all: for any square matrix $A$,

where $I$ is the identity matrix of the same order. Why the diagonal entries work out. Each diagonal entry of $A(\operatorname{adj}A)$ is a row of $A$ dotted with the matching column of $\operatorname{adj}A$ -- but that column of $\operatorname{adj}A$ is exactly the cofactors $C_{i1},C_{i2},C_{i3}$ of row $i$ (transposed into a column), so the dot product is precisely $a_{i1}C_{i1}+a_{i2}C_{i2}+a_{i3}C_{i3}=|A|$, the expansion-along-row-$i$ formula from Section 3. Why the off-diagonal entries vanish. Every off-diagonal entry pairs a row of $A$ with the cofactors of a different row -- and it is a standard (if less obvious) fact that such a "mismatched" sum always equals $0$, because it is exactly the expansion of a determinant with two identical rows (Property 3, Section 2), which is always zero. …