Question

If the system of equations $3x-2y+z=0$, $λx-14y+15z=0$, $x+2y+3z=0$ have a non-trivial solution, then $λ=$

• A. 29
• B. -15
• C. -29
• D. 15

Answer 1

Answer: (A)

29

Answer 2

For a homogeneous system of linear equations to have a non-trivial solution, the determinant of its coefficient matrix must be zero. We form the coefficient matrix from the given equations:

$A = \begin{pmatrix} 3 & -2 & 1 \\ λ & -14 & 15 \\ 1 & 2 & 3 \end{pmatrix}$

Calculate the determinant:

$\det(A) = 3 \begin{vmatrix} -14 & 15 \\ 2 & 3 \end{vmatrix} - (-2) \begin{vmatrix} λ & 15 \\ 1 & 3 \end{vmatrix} + 1 \begin{vmatrix} λ & -14 \\ 1 & 2 \end{vmatrix}$

$\det(A) = 3 ((-14)(3) - (15)(2)) + 2 ((λ)(3) - (15)(1)) + 1 ((λ)(2) - (-14)(1))$

$\det(A) = 3 (-42 - 30) + 2 (3λ - 15) + 1 (2λ + 14)$

$\det(A) = 3 (-72) + 6λ - 30 + 2λ + 14$

$\det(A) = -216 + 8λ - 16$

$\det(A) = 8λ - 232$

Set $\det(A) = 0$:

$8λ - 232 = 0$ $8λ = 232$ $λ = \frac{232}{8}$ $λ = 29$