[latex]
Question #1:
\[
\textbf{Calculate the determinant of the following upper triangular matrix:}
\]
\[
A = \begin{pmatrix} 4 & 3 & 2 \\ 0 & 5 & 1 \\ 0 & 0 & 6 \end{pmatrix}
\]
Question #2:
\[
\textbf{ Calculate the determinant of the following lower triangular matrix:}
\]
\[
B = \begin{pmatrix} 2 & 0 & 0 \\ 3 & 4 & 0 \\ 1 & 6 & 5 \end{pmatrix}
\]
Question #3:
\[
\textbf{ Calculate the determinant of the following upper triangular matrix:}
\]
\[
C = \begin{pmatrix} 7 & 2 & 1 \\ 0 & 3 & 4 \\ 0 & 0 & 9 \end{pmatrix}
\]
Question #4:
\[
\textbf{Calculate the determinant of the following lower triangular matrix:}
\]
\[
D = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 8 \end{pmatrix}
\]
Solutions
\[
\text{Recall: The determinant of a triangular matrix is the product of the diagonal elements.}
\]
Question #1:
\[
\textbf{ Calculate the determinant of the following upper triangular matrix:}
\]
\[
A = \begin{pmatrix} 4 & 3 & 2 \\ 0 & 5 & 1 \\ 0 & 0 & 6 \end{pmatrix}
\]
\[
\text{Since this is an upper triangular matrix, we calculate the determinant by multiplying the diagonal elements:}
\]
\[
\text{det}(A) = 4 \times 5 \times 6 = 120
\]
\[
\textbf{Answer: The determinant of matrix } A \text{ is } 120.
\]
Question #2:
\[
\textbf{ Calculate the determinant of the following lower triangular matrix:}
\]
\[
B = \begin{pmatrix} 2 & 0 & 0 \\ 3 & 4 & 0 \\ 1 & 6 & 5 \end{pmatrix}
\]
\[
\text{Since this is a lower triangular matrix, we calculate the determinant by multiplying the diagonal elements:}
\]
\[
\text{det}(B) = 2 \times 4 \times 5 = 40
\]
\[
\textbf{Answer: The determinant of matrix } B \text{ is } 40.
\]
Question #3:
\[
\textbf{ Calculate the determinant of the following upper triangular matrix:}
\]
\[
C = \begin{pmatrix} 7 & 2 & 1 \\ 0 & 3 & 4 \\ 0 & 0 & 9 \end{pmatrix}
\]
\[
\text{Since this is an upper triangular matrix, we calculate the determinant by multiplying the diagonal elements:}
\]
\[
\text{det}(C) = 7 \times 3 \times 9 = 189
\]
\[
\textbf{Answer: The determinant of matrix } C \text{ is } 189.
\]
Question #4:
\[
\textbf{Calculate the determinant of the following lower triangular matrix:}
\]
\[
D = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 8 \end{pmatrix}
\]
\[
\text{Since this is a lower triangular matrix, we calculate the determinant by multiplying the diagonal elements:}
\]
\[
\text{det}(D) = 1 \times (-3) \times 8 = -24
\]
\[
\textbf{Answer: The determinant of matrix } D \text{ is } -24.
\]