Question

If \[ \begin{vmatrix} -1 & -2 & 5 \\ -2 & a & 4 \\ 0 & 4 & 2a \end{vmatrix} = -86 \] then the sum of all possible values of \( a \) is

• A. 4
• B. 5
• C. $-4$
• D. 9

Hint:

For quadratic equations \(ax^2+bx+c=0\), sum of roots = \(-\frac{b}{a}\).

Answer 1

The Correct Option is A

Step 1: Expand the determinant.
Given determinant
\[ \begin{vmatrix} -1 & -2 & 5 \\ -2 & a & 4 \\ 0 & 4 & 2a \end{vmatrix} \] Expand along the first row.
\[ = -1 \begin{vmatrix} a & 4 \\ 4 & 2a \end{vmatrix} -(-2) \begin{vmatrix} -2 & 4 \\ 0 & 2a \end{vmatrix} +5 \begin{vmatrix} -2 & a \\ 0 & 4 \end{vmatrix} \] Step 2: Compute each minor determinant.
First minor
\[ \begin{vmatrix} a & 4 \\ 4 & 2a \end{vmatrix} = 2a^2 - 16 \] Second minor
\[ \begin{vmatrix} -2 & 4 \\ 0 & 2a \end{vmatrix} = -4a \] Third minor
\[ \begin{vmatrix} -2 & a \\ 0 & 4 \end{vmatrix} = -8 \] Step 3: Substitute into expansion.
\[ = -1(2a^2 - 16) + 2(-4a) + 5(-8) \] \[ = -2a^2 + 16 - 8a - 40 \] \[ = -2a^2 - 8a - 24 \] Step 4: Use the given condition.
Given determinant
\[ = -86 \] Thus
\[ -2a^2 - 8a - 24 = -86 \] \[ -2a^2 - 8a + 62 = 0 \] Divide by \(-2\)
\[ a^2 + 4a - 31 = 0 \] Step 5: Solve quadratic equation.
Using quadratic formula
\[ a = \frac{-4 \pm \sqrt{16 + 124}}{2} \] \[ a = \frac{-4 \pm \sqrt{140}}{2} \] Thus two values exist.
Step 6: Find sum of roots.
For quadratic
\[ ax^2 + bx + c = 0 \] Sum of roots
\[ = -\frac{b}{a} \] Thus
\[ = -\frac{4}{1} \] \[ = -4 \] Step 7: Conclusion.
The sum of all possible values of \( a \) is
\[ -4 \] Final Answer: \( \boxed{-4} \)