Step 1 (Formula / Governing Law & Given Data): [0.25 Mark]

The determinant of a $3 \times 3$ matrix $A = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$ is calculated by expanding along any row or column. Given the determinant equation:

$$\begin{vmatrix} -1 & -2 & 5 \\ -2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = -86$$

Step 2 (Step-by-Step Mathematical Working): [0.5 Mark]

Expanding along the first column ($C_1$) to simplify the calculation due to the zero element:

$$-1 \begin{vmatrix} a & -1 \\ 4 & 2a \end{vmatrix} - (-2) \begin{vmatrix} -2 & 5 \\ 4 & 2a \end{vmatrix} + 0 \begin{vmatrix} -2 & 5 \\ a & -1 \end{vmatrix} = -86$$

Calculating the $2 \times 2$ determinants:

$$-1(2a^2 - (-4)) + 2(-4a - 20) = -86$$ $$-1(2a^2 + 4) - 8a - 40 = -86$$ $$-2a^2 - 4 - 8a - 40 = -86$$ $$-2a^2 - 8a - 44 = -86$$

Rearranging into a standard quadratic equation $ax^2 + bx + c = 0$:

$$-2a^2 - 8a + 42 = 0$$

Dividing by $-2$:

$$a^2 + 4a - 21 = 0$$

Step 3 (Final Concluded Answer): [0.25 Mark]

For a quadratic equation $ax^2 + bx + c = 0$, the sum of roots is given by $-\frac{b}{a}$. Here, $a=1$ and $b=4$.

$$\text{Sum of roots} = -\frac{4}{1} = -4$$

Alternatively, factoring gives $(a+7)(a-3) = 0$, so $a = -7$ and $a = 3$. The sum is $-7 + 3 = -4$.

Correct Option: (C) $-4$

💡 EL10 Concept Insight: Think of a determinant as a way to measure the "area" or "volume" created by the numbers in the grid; when we expand it, we are just breaking that big volume into smaller, manageable pieces to solve for the unknown variable $a$. Since the equation turned into a quadratic, we simply used the shortcut that the sum of the answers is always the middle number with its sign flipped!