1. **Problem:** Find the Adjoint Matrix of the given matrix $$D = \begin{bmatrix} 11 & 9 & -4 \\ -2 & 0 & 1 \\ 0 & -6 & 1 \end{bmatrix}$$. 2. **Formula and Explanation:** The adjoint (or adjugate) matrix of a square matrix is the transpose of the cofactor matrix. - Step 1: Find the cofactor matrix by calculating the cofactor of each element. - Step 2: Transpose the cofactor matrix to get the adjoint matrix. 3. **Calculate Cofactors:** - Cofactor $$C_{11}$$: Minor of element at row 1, column 1 is $$\begin{vmatrix} 0 & 1 \\ -6 & 1 \end{vmatrix} = (0)(1) - (1)(-6) = 6$$, so $$C_{11} = (+1) \times 6 = 6$$. - Cofactor $$C_{12}$$: Minor of element at row 1, column 2 is $$\begin{vmatrix} -2 & 1 \\ 0 & 1 \end{vmatrix} = (-2)(1) - (1)(0) = -2$$, so $$C_{12} = (-1) \times -2 = 2$$. - Cofactor $$C_{13}$$: Minor of element at row 1, column 3 is $$\begin{vmatrix} -2 & 0 \\ 0 & -6 \end{vmatrix} = (-2)(-6) - (0)(0) = 12$$, so $$C_{13} = (+1) \times 12 = 12$$. - Cofactor $$C_{21}$$: Minor of element at row 2, column 1 is $$\begin{vmatrix} 9 & -4 \\ -6 & 1 \end{vmatrix} = (9)(1) - (-4)(-6) = 9 - 24 = -15$$, so $$C_{21} = (-1) \times -15 = 15$$. - Cofactor $$C_{22}$$: Minor of element at row 2, column 2 is $$\begin{vmatrix} 11 & -4 \\ 0 & 1 \end{vmatrix} = (11)(1) - (-4)(0) = 11$$, so $$C_{22} = (+1) \times 11 = 11$$. - Cofactor $$C_{23}$$: Minor of element at row 2, column 3 is $$\begin{vmatrix} 11 & 9 \\ 0 & -6 \end{vmatrix} = (11)(-6) - (9)(0) = -66$$, so $$C_{23} = (-1) \times -66 = 66$$. - Cofactor $$C_{31}$$: Minor of element at row 3, column 1 is $$\begin{vmatrix} 9 & -4 \\ 0 & 1 \end{vmatrix} = (9)(1) - (-4)(0) = 9$$, so $$C_{31} = (+1) \times 9 = 9$$. - Cofactor $$C_{32}$$: Minor of element at row 3, column 2 is $$\begin{vmatrix} 11 & -4 \\ -2 & 1 \end{vmatrix} = (11)(1) - (-4)(-2) = 11 - 8 = 3$$, so $$C_{32} = (-1) \times 3 = -3$$. - Cofactor $$C_{33}$$: Minor of element at row 3, column 3 is $$\begin{vmatrix} 11 & 9 \\ -2 & 0 \end{vmatrix} = (11)(0) - (9)(-2) = 18$$, so $$C_{33} = (+1) \times 18 = 18$$. 4. **Cofactor Matrix:** $$\begin{bmatrix} 6 & 2 & 12 \\ 15 & 11 & 66 \\ 9 & -3 & 18 \end{bmatrix}$$ 5. **Adjoint Matrix:** Transpose the cofactor matrix: $$\text{Adj}(D) = \begin{bmatrix} 6 & 15 & 9 \\ 2 & 11 & -3 \\ 12 & 66 & 18 \end{bmatrix}$$ **Final answer:** $$\boxed{\text{Adj}(D) = \begin{bmatrix} 6 & 15 & 9 \\ 2 & 11 & -3 \\ 12 & 66 & 18 \end{bmatrix}}$$