1. **Problem:** Find the determinant of the matrix $$\begin{bmatrix}-3 & 0 & 0 \\ 7 & 11 & 0 \\ 1 & 2 & 2\end{bmatrix}$$ using expansion by cofactors. 2. **Formula and rules:** The determinant of a 3x3 matrix $$A = \begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}$$ is $$\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg).$$ Expansion by cofactors means choosing a row or column and summing the products of each element, its cofactor sign, and the determinant of the minor matrix. 3. **Step (a): Expansion by the third row** The third row is $[1, 2, 2]$. Cofactor signs for the third row are $[+, -, +]$. Calculate minors: - Minor for element $1$ (position (3,1)) is determinant of $$\begin{bmatrix}0 & 0 \\ 11 & 0\end{bmatrix} = 0 \times 0 - 0 \times 11 = 0.$$ - Minor for element $2$ (position (3,2)) is determinant of $$\begin{bmatrix}-3 & 0 \\ 7 & 0\end{bmatrix} = (-3) \times 0 - 0 \times 7 = 0.$$ - Minor for element $2$ (position (3,3)) is determinant of $$\begin{bmatrix}-3 & 0 \\ 7 & 11\end{bmatrix} = (-3) \times 11 - 0 \times 7 = -33.$$ Calculate determinant: $$\det = (+)1 \times 0 - 2 \times 0 + 2 \times (-33) = 0 - 0 - 66 = -66.$$ 4. **Step (b): Expansion by the first column** The first column is $[-3, 7, 1]$. Cofactor signs for the first column are $[+, -, +]$. Calculate minors: - Minor for element $-3$ (position (1,1)) is determinant of $$\begin{bmatrix}11 & 0 \\ 2 & 2\end{bmatrix} = 11 \times 2 - 0 \times 2 = 22.$$ - Minor for element $7$ (position (2,1)) is determinant of $$\begin{bmatrix}0 & 0 \\ 2 & 2\end{bmatrix} = 0 \times 2 - 0 \times 2 = 0.$$ - Minor for element $1$ (position (3,1)) is determinant of $$\begin{bmatrix}0 & 0 \\ 11 & 0\end{bmatrix} = 0 \times 0 - 0 \times 11 = 0.$$ Calculate determinant: $$\det = (+)(-3) \times 22 - 7 \times 0 + 1 \times 0 = -66 - 0 + 0 = -66.$$ **Final answer:** $$\boxed{-66}$$