First slide

Applications of determinant

Question

Let $ω \neq 1$ be a cube root of unity and S be the set of all non-singular matrices of the form $A = [\begin{array}{ccc} 1 & a & b \\ ω & 1 & c \\ ω^{2} & ω & 1 \end{array}]$
where $a, b, c$ are either $ω$ or $ω^{2}$ . Then number of distinct matrices in the set S is

Moderate

A

2
B

6
C

4
D

8

Solution

We have
$\begin{aligned} | A | = 1 |\begin{array}{ll} 1 & c \\ ω & 1 \end{array}| - a |\begin{array}{cc} ω & c \\ ω^{2} & 1 \end{array}| + b |\begin{array}{cc} ω & 1 \\ ω^{2} & ω \end{array}| \\ = 1 - c ω - a ω + a c ω^{2} = (1 - a ω) (1 - c ω) \end{aligned}$
A is non-singular $\Leftrightarrow | A | \neq 0 \Leftrightarrow a = c = ω .$
Note that b can take any value $ω$ or $ω^{2}$ .
Thus, there are two non-singular matrices.

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