Question

If the systems of equations $3x - 2y + z = 0$, $5x + ay + 15z = 0$, $x + 2y - 3z = 0$ have non-zero solution, then $a =$ ...............

• A. $-2$
• B. $2$
• C. $-14$
• D. $14$

Hint:

For homogeneous systems, a non-zero solution exists only when the determinant of the coefficient matrix is zero.

Answer 1

The Correct Option is C

We are given a system of 3 linear homogeneous equations in 3 variables.

To have a non-zero solution, the determinant of the coefficient matrix must be zero.

Let the coefficient matrix be:
\[ A = \begin{bmatrix} 3 & -2 & 1 \\ 5 & a & 15 \\ 1 & 2 & -3 \end{bmatrix} \]

Compute the determinant of \( A \):
\[ \text{Det}(A) = 3 \begin{vmatrix} a & 15 \\ 2 & -3 \end{vmatrix} - (-2) \begin{vmatrix} 5 & 15 \\ 1 & -3 \end{vmatrix} + 1 \begin{vmatrix} 5 & a \\ 1 & 2 \end{vmatrix} \]

Calculate each minor:
\[ = 3 (a \cdot (-3) - 15 \cdot 2) + 2 (5 \cdot (-3) - 15 \cdot 1) + 1 (5 \cdot 2 - a \cdot 1) \]
\[ = 3(-3a - 30) + 2(-15 - 15) + (10 - a) = -9a - 90 - 60 + 10 - a = -10a - 140 \]

Set determinant equal to zero for non-trivial solution:
\[ -10a - 140 = 0 \Rightarrow -10a = 140 \Rightarrow a = -14 \]