First slide

System of linear equations

Question

Let $λ$ and $α$ be real. Then the number of integral values $λ$ for which the system of linear equations
$\begin{array}{l} λx + (\sin α) y + (\cos α) z = 0; \\ x + (\cos α) y + (\sin α) z = 0; \\ - x + (\sin α) y - (\cos α) z = 0 \end{array}$
has non-trivial solutions is

Moderate

A

0
B

1
C

2
D

3

Solution

The given system has a non-trivial solution if
$|\begin{array}{ccc} λ & \sin α & \cos α \\ 1 & \cos α & \sin α \\ - 1 & \sin α & - \cos α \end{array}| = 0$
By expanding the determinant along first column, we get
$λ = \sin 2 α + \cos 2 α$
We know that
$- \sqrt{2} \leq \sin 2 α + \cos 2 α \leq \sqrt{2}$
$∴ - \sqrt{2} \leq λ \leq \sqrt{2}$
Hence, integral values of $λ$ are -1, 0 and 1.

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