First slide

System of linear equations

Question

The set of equations: $λ x - y + (\cos θ) z = 0; 3 x + y + 2 z = 0;$ $(\cos θ) x + y + 2 z = 0$ $0 \leq θ < 2 π,$ has non-trivial solutions

Moderate

A

For no value of $λ a n d θ$
B

For all values of $λ a n d θ$
C

For all values of $λ$ and only two values of $θ$
D

For only one value of $λ$ and all values of $θ$

Solution

Determinant of coefficients $= | \begin{matrix} λ - 1 \cos θ \\ 3 1 2 \\ \cos θ 1 2 \end{matrix} | = \cos θ - \cos^{2} θ + 6$ and this is positive for all $θ$ since $| \cos θ | \leq 1$
The only solution is therefore the trivial solution.

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