Let ω ≠ 1 be a cube root of unity and S be the set of all non‑singular matrices of the form \left[\right. 1 & a & b \\ \omega & 1 & c \\ \left(\omega\right)^{2} & \omega & 1 \left]\right. where each of a, b and c is either ω or ω2. Then the number of distinct matrices in the set S is

Engineering

Mathematics

Cube Roots of Unity

Question

Let ω ≠ 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 ω². Then the number of distinct matrices in the set S is

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

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

(1 – $ω$ c) – a ( $ω$ – $ω$ ²c) + b( $ω$ ² – $ω$ ²) $\neq$ 0

1 – $ω$ c – a $ω$ + ac $ω$ ² $\neq$ 0

(1 – $ω$ c) – a $ω$ (1 – $ω$ c) $\neq$ 0

(1 – $ω$ c) (1 – a $ω$ ) $\neq$ 0

c $\neq$ $ω$ ² & a $\neq ω$ ² & b = $ω or ω$ ²

(a, b, c) $\equiv (ω, ω, ω) or (ω, ω^{2}, ω)$

Please subscribe our Youtube channel to unlock this solution.