Let ω ≠ 1 be a cube root of unity and S be the set of all non-singular matrices of the form . 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

Introduction to Determinants

Question

Let ω ≠ 1 be a cube root of unity and S be the set of all non-singular matrices of the form $[\begin{array}{ccc} 1 & a & b \\ ω & 1 & c \\ ω^{2} & ω & 1 \end{array}]$ . Where each of a, b and c is either ω or ω². Then the number of distinct matrices in the set S is

LiveFree JEE Main Previous Year Online Papers Live Free JEE Advance Previous Year Online Papers

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

For the given matrix to be non-singular

1 – (a + c)w + acw² ≠ 0

(1 – aw) (1 – cw) ≠ 0

a ≠ w² and c ≠ w²

where w is complex cube root of unity

And a, b, c are complex cube root of unity

therefore, a and c can take two values w and w²

Hence, total numbers of distinct matrices

= 1 × 1 × 2 = 2

Please subscribe our Youtube channel to unlock this solution.