Adjoint of a square matrix
- To find the adjoint of the a square matrix A:
- We form the matrix C of cofactors.
- We write the transpose of C, i.e. CT
Therefore:
\[\Large
\textcolor{red}{\text{adj}(A) = C^T}
\]
Example:
\[\Large
A = \begin{bmatrix}
2 & 3 & 5 \\
4 & 1 & 6 \\
1 & 5 & 0
\end{bmatrix}
\]
\[\Large C = \begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix} \]
\[\Large C = \begin{bmatrix} -24 & 6 & 15 \\ 20 & -5 & -5 \\ 13 & 8 & -10 \end{bmatrix} \]
\[\Large C^T = \begin{bmatrix} -24 & 20 & 13 \\ 6 & -5 & 8 \\ 15 & -5 & -10 \end{bmatrix} \]
\[\Large adj(A)=\begin{bmatrix} -24 & 20 & 13 \\ 6 & -5 & 8 \\ 15 & -5 & -10 \end{bmatrix} \]
\[\Large C = \begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix} \]
\[\Large C = \begin{bmatrix} -24 & 6 & 15 \\ 20 & -5 & -5 \\ 13 & 8 & -10 \end{bmatrix} \]
\[\Large C^T = \begin{bmatrix} -24 & 20 & 13 \\ 6 & -5 & 8 \\ 15 & -5 & -10 \end{bmatrix} \]
\[\Large adj(A)=\begin{bmatrix} -24 & 20 & 13 \\ 6 & -5 & 8 \\ 15 & -5 & -10 \end{bmatrix} \]