Find the adjoint of the matrix \( \left[\begin{array}{cc}2 & -4 \\ -1 & 2\end{array}\right] \)

To find the adjoint of the matrix \( A = \left[\begin{array}{cc}2 & -4 \\ -1 & 2\end{array}\right] \), we first need to compute the cofactor matrix. The adjoint of a matrix is the transpose of its cofactor matrix. 1. The matrix \( A \) is a \( 2 \times 2 \) matrix, and its minors can be calculated as follows: - The minor of the element at position (1,1) is \( 2 \) (from the submatrix \( \left[\begin{array}{cc}2\end{array}\right] \)). - The minor of the element at position (1,2) is \( -1 \) (from the submatrix \( \left[\begin{array}{cc}-1\end{array}\right] \)). - The minor of the element at position (2,1) is \( -4 \). - The minor of the element at position (2,2) is \( 2 \). 2. Next, we calculate the cofactors, which also involve applying the checkerboard pattern of signs: - \( C_{11} = M_{11} = 2 \) - \( C_{12} = -M_{12} = 1 \) (note the negative sign) - \( C_{21} = -M_{21} = 4 \) (note the negative sign) - \( C_{22} = M_{22} = 2 \) 3. The cofactor matrix \( C \) will be: \[ C = \left[\begin{array}{cc} 2 & 1 \\ 4 & 2 \end{array}\right] \] 4. The adjoint (or adjugate) of the matrix \( A \) is obtained by taking the transpose of the cofactor matrix \( C \): \[ \text{Adj}(A) = C^T = \left[\begin{array}{cc} 2 & 4 \\ 1 & 2 \end{array}\right] \] Thus, the adjoint of the matrix \( \left[\begin{array}{cc}2 & -4 \\ -1 & 2\end{array}\right] \) is \[ \boxed{\left[\begin{array}{cc} 2 & 4 \\ 1 & 2 \end{array}\right]} \]