Find the determinant of $\left[ \begin{array} { c c c } 0 & - 6 & 0 \ - 9 & 6 & - 18 \ 3 & 0 & 18 \end{array} \right]$ by the method of expansion by cofactors.
A) $- 648$
B) 1296
C) $- 1296$
D) 162
E) $- 594$

Question

Quizplus · Accepted Answer

The determinant of a 3x3 matrix $\left[ \begin{array} { c c c } a & b & c \ d & e & f \ g & h & i \end{array} \right]$ can be calculated by the method of expansion by cofactors along any row or column. Choosing the first row for simplicity, because it contains two zeros, the determinant is given by:$$0 \cdot C_{11} - 6 \cdot C_{12} + 0 \cdot C_{13}$$where $C_{ij}$ is the cofactor of the element in the $i$th row and $j$th column. The cofactor $C_{ij}$ is calculated as $(-1)^{i+j} \cdot M_{ij}$, where $M_{ij}$ is the determinant of the matrix that remains after removing the $i$th row and $j$th column from the original matrix.For the given matrix, the cofactor $C_{12}$ is needed, which corresponds to removing the first row and the second column, resulting in the 2x2 matrix $\left[ \begin{array} { c c } -9 & -18 \ 3 & 18 \end{array} \right]$. The determinant of this 2x2 matrix is $(-9) \cdot 18 - (-18) \cdot 3 = -162 + 54 = -108$.Therefore, the determinant of the original matrix is:$$-6 \cdot (-1)^{1+2} \cdot (-108) = -6 \cdot (-108) = 648$$So, the correct answer is $-648$, which is choice A.

Find the Determinant Of [0−60−96−183018]\left[ \begin{array} { c c c } 0 & - 6 & 0 \\ - 9 & 6 & - 18 \\ 3 & 0 & 18 \end{array} \right]​0−93​−660​0−1818​​

Find the Determinant Of $\left[ \begin{array} { c c c } 0 & - 6 & 0 \\ - 9 & 6 & - 18 \\ 3 & 0 & 18 \end{array} \right]$