First slide

System of linear equations

Question

If the system of equations
$\begin{array}{l} (a - t) x + by + cz = 0 \\ bx + (c - t) y + az = 0 \\ cx + ay + (b - t) z = 0 \end{array}$
has non-trivial solution, then product of all possible values of t is

Moderate

A

$|\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}|$
B

a + b + c
C

a² + b² + c²
D

1

Solution

The given system of equations will have a non-trivial solution if the determinant of coefficients is 0.
$Δ = |\begin{array}{ccc} a - t & b & c \\ b & c - t & a \\ c & a & b - t \end{array}| = 0$ ………. (1)
$∆ = 0$ is a cubic equation in t, so it has 3 solutions say t₁, t₂ and t₃.
Let $Δ = p_{0} t^{3} + p_{1} t^{2} + p_{2} t + p_{3}$ ,
Clearly, p_o= coeff. of t³ which is equal to -1, so
$t_{1} t_{2} t_{3} = - \frac{p_{3}}{(- 1)} = p_{3}$
= Constant term in the expansion of $Δ i.e. Δ_{(t = 0)}$ .
Hence, $t_{1} t_{2} t_{3} = |\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}|$

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