Question

Find the adjoint of the following matrix:
\[\begin{bmatrix}a & b \\ c & d\end{bmatrix}\]

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.

Answer 1

Given below is the square matrix. Here, we will interchange the diagonal elements and change the signs of the off-diagonal elements.
\[\ B = \begin{bmatrix}a & b \\ c & d\end{bmatrix}\]
\[adjB = \begin{bmatrix}d & - b \\ - c & a\end{bmatrix}\]
\[(adjB)B = \begin{bmatrix}ad - bc & 0 \\ 0 & - cb + ad\end{bmatrix}\]
\[\left| B \right| = ad - bc\]
\[\left| B \right|I = \begin{bmatrix}ad - bc & 0 \\ 0 & - cb + ad\end{bmatrix}\]
\[B(adjB) = \begin{bmatrix}ad - bc & 0 \\ 0 & - cb + ad\end{bmatrix}\]
\[ \therefore (adjB)B = \left| B \right|I = B(adjB)\]
Hence verified.