Find the determinant of the matrix by the method of expansion by cofactors.Expand using the column 2. $$\left[ \begin{array} { c c c }
- 3 & 2 & 1 \
4 & 5 & 6 \
2 & - 3 & 1
\end{array} \right]$$
A)-75
B)75
C)-73
D)-74
E)-76

Question

Find the determinant of the matrix by the method of expansion by cofactors.Expand using the column 2.​ $$\left[ \begin{array} { c c c } 
- 3 & 2 & 1 \
4 & 5 & 6 \
2 & - 3 & 1
\end{array} \right]$$ ​
A)-75
B)75
C)-73
D)-74
E)-76

Quizplus · Accepted Answer

Expanding the determinant along column 2, we get: $2(-1)^{1+2}(4*1 - 6*2) + 5(-1)^{2+2}( -3*1 - 1*2) + (-3)(-1)^{3+2}( -3*6 - 4*1)$. Simplifying, we get: $2(-14) + 5(-5) - 3(-22) = -28 - 25 + 66 = 13$. However, there seems to be a mistake in my calculation as none of the options match the result. Let's correct the calculation: The correct expansion is: $2(-1)^{1+2}(4*1 - 6*2) + 5(-1)^{2+2}( -3*1 - 1*2) + (-3)(-1)^{3+2}( -3*6 - 4*1)$ = $2(-8) + 5(-5) - 3(-22)$ = $-16 - 25 + 66 = 25$. Again, my recalculation seems incorrect as it does not match the provided options, indicating a mistake in my process.Correcting the mistake and properly calculating the determinant by expanding along column 2: The determinant is $2(-1)^{1+2}(4*1 - 6*2) + 5(-1)^{2+2}(-3*1 - 2*1) + (-3)(-1)^{3+2}( -3*6 - 4*1)$ = $2(-8) + 5(-5) + (-3)(-22)$ = $-16 - 25 - 66$, which still does not align with the correct approach or answer. The correct approach is to carefully apply the method of cofactors for each element in column 2, ensuring accurate calculation of minors and cofactors. My explanations contained errors in calculation and did not accurately follow through the correct method of cofactor expansion. The correct calculation should directly apply the formula for expansion by cofactors along the chosen column, accurately computing each minor and its sign change according to the position in the matrix.

Find the Determinant of the Matrix by the Method of Expansion