Choose the correct option for the following determinant

Question

Do we have to expand the determinant to find sum of Coefficients or coefficient of any power of $x$ or can it be calculated without expanding too?

Answer 1

The coefficient $a_7$ is $\Delta(0)$, so $$ \det\begin{bmatrix} 0 & -1 & 3 \\ 1 & 2 & -3 \\ -3 & 4 & 0 \end{bmatrix}=21 $$

Similarly, for $\Delta(1)$ substitute $x=1$: $$ \Delta(1)=\det\begin{bmatrix} 2 & 1 & 4 \\ 4 & 3 & -2 \\ -2 & 5 & 2 \end{bmatrix}=132 $$

You also have $a_0+a_1+a_2+a_3+a_4+a_5+a_6+a_7=\Delta(1)$, so $$ a_0+a_1+a_2+a_3+a_4+a_5+a_6=\Delta(1)-a_7=111 $$

You can do $\Delta(-1)$ yourself (it gives $-32$).

Answer 2

Forget about all but the constant terms for each element in order to simplify the calculation of $a_7$.

Similarly you can forget about all but the leading terms for each element in order to calculate $a_0$. Once you have this, you can subtract it from $\Delta(1)$ to obtain $\sum_{k=0}^6 a_k$.