Steps to Find the Adjoint of a Square Matrix
- Steps
- Calculate cofactors for each element of A.
- Form the cofactor matrix.
- Transpose the cofactor matrix to get the Adjoint of A.
- Example:
\[\Large
A =
\begin{bmatrix}
2 & 1 & 4 \\
1 & 6 & 3 \\
5 & 2 & 2
\end{bmatrix}
\]
- Let us find the cofactor Matrix for A.
\[\Large
C =
\begin{bmatrix}
\color{red}{-24} & \color{green}{6} & \color{blue}{15} \\
\color{red}{-20} & \color{green}{5} & \color{blue}{-5} \\
\color{red}{13} & \color{green}{8} & \color{blue}{-10}
\end{bmatrix}
\]
- Let us find the Transpose of C
\[\Large
C^T =
\begin{bmatrix}
\color{red}{-24} & \color{red}{-20} & \color{red}{13} \\
\color{green}{6} & \color{green}{5} & \color{green}{8} \\
\color{blue}{15} & \color{blue}{-5} & \color{blue}{-10}
\end{bmatrix}
\]
\[\Large \text{Since } \text{adj}(A) = C^T, \text{ therefore, } \text{adj}(A) = \begin{bmatrix} \color{red}{-24} & \color{red}{-20} & \color{red}{13} \\ \color{green}{6} & \color{green}{5} & \color{green}{8} \\ \color{blue}{15} & \color{blue}{-5} & \color{blue}{-10} \end{bmatrix}\]
Which of the following is the correct step to compute the adjoint of a square matrix?