Let be a cube root of unity and be the set of all non-singular matrices of the form where each of , and is either or . Then the number of distinct matrices in the set is [2011]

Question

Let $ω \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $[\begin{matrix} 1 & a & b \\ ω & 1 & c \\ ω^{2} & ω & 1 \end{matrix}]$ where each of $a$ , $b$ and $c$ is either $ω$ or $ω^{2}$ . Then the number of distinct matrices in the set $S$ is [2011]

Smart Achievers · Accepted Answer

(1)

For the given matrix to be non-singular

$| \begin{matrix} 1 & a & b \\ ω & 1 & c \\ ω^{2} & ω & 1 \end{matrix} |$ $\neq 0$

$a$ $\Rightarrow (1 - a ω) (1 - c ω) \neq 0$

$\Rightarrow a \neq ω^{2} and c \neq ω^{2},$ where $ω$ is a complex cube root of unity.

As $a$ , $b$ and $c$ are complex cube roots of unity,

$∴$ $a$ and $c$ can take only one value, i.e. $ω$ , while $b$ can take two values, i.e. $ω$ and $ω^{2}$ .

$∴$ Total number of distinct matrices in the set S $= 1 \times 1 \times 2 = 2$

Solution

Q.
Let $ω \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $[\begin{matrix} 1 & a & b \\ ω & 1 & c \\ ω^{2} & ω & 1 \end{matrix}]$ where each of $a$ , $b$ and $c$ is either $ω$ or $ω^{2}$ . Then the number of distinct matrices in the set $S$ is [2011]

1 2

2 6

3 4

4 8

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Solution

Q. Let ω≠1 be a cube root of unity and S be the set of all non-singular matrices of the form [1abω1cω2ω1] where each of a, b and c is either ω or ω2. Then the number of distinct matrices in the set S is [2011]

1 2

2 6

3 4

4 8

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