Question

Find the adjoint of the matrix \( \left[ \begin{matrix} 2 & -2 \\ 4 & 3 \end{matrix} \right] \).

Hint:

To find the adjoint, first find the cofactor matrix and then transpose it.

Answer 1

Step 1: Recall the formula for adjoint of a matrix. \\ The adjoint of a matrix is the transpose of the cofactor matrix. For a 2x2 matrix \( A = \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \), the cofactor matrix is given by: \[ \text{Cofactor}(A) = \left[ \begin{matrix} d & -b \\ -c & a \end{matrix} \right] \] Step 2: Calculate the cofactors for the matrix. \\ For the matrix \( \left[ \begin{matrix} 2 & -2 \\ 4 & 3 \end{matrix} \right] \), the cofactor matrix is: \[ \text{Cofactor}(A) = \left[ \begin{matrix} 3 & 2 \\ -4 & 2 \end{matrix} \right] \] Step 3: Find the adjoint by transposing the cofactor matrix. \\ The adjoint is the transpose of the cofactor matrix: \[ \text{Adjoint}(A) = \left[ \begin{matrix} 3 & -4 \\ 2 & 2 \end{matrix} \right] \]

Final Answer: \[ \boxed{\left[ \begin{matrix} 3 & -4 \\ 2 & 2 \end{matrix} \right]} \]