Question

Find the adjoint of the matrix: \[ \left[ \begin{matrix} 2 & 3 \\ 5 & 4 \end{matrix} \right] \]

• A. \( \left[ \begin{matrix} 4 & -5 \\ -3 & 2 \end{matrix} \right]\)
• B. \( \left[ \begin{matrix} 4 & -3 \\ -5 & 2 \end{matrix} \right]\)
• C. \( \left[ \begin{matrix} 4 & 5 \\ 3 & 2 \end{matrix} \right]\)
• D. \( \left[ \begin{matrix} 4 & 3 \\ 5 & 2 \end{matrix} \right] \)

Answer 1

The Correct Option is B

To find the adjoint of a matrix, we first calculate the cofactor matrix and then transpose it. Given matrix \( A = \left[ \begin{matrix} 2 & 3 \\ 5 & 4 \end{matrix} \right] \), we will find the cofactor matrix by finding the minors and then the cofactors: - For the (1,1) element: The minor is the determinant of \( \left[ \begin{matrix} 4 \end{matrix} \right] \), which is 4. The cofactor is \( +4 \). - For the (1,2) element: The minor is the determinant of \( \left[ \begin{matrix} 5 \end{matrix} \right] \), which is 5. The cofactor is \( -5 \). - For the (2,1) element: The minor is the determinant of \( \left[ \begin{matrix} 3 \end{matrix} \right] \), which is 3. The cofactor is \( -3 \). - For the (2,2) element: The minor is the determinant of \( \left[ \begin{matrix} 2 \end{matrix} \right] \), which is 2. The cofactor is \( +2 \). Thus, the cofactor matrix is: \[ \left[ \begin{matrix} 4 & -5 \\ -3 & 2 \end{matrix} \right] \] Now, we take the transpose of the cofactor matrix to get the adjoint matrix, which remains the same in this case. Therefore, the adjoint is: \[ \left[ \begin{matrix} 4 & -3 \\ -5 & 2 \end{matrix} \right] \]