Determinants cube root of unity question: Evaluate: | 1 w3 w5 | |w3 1 - Maths - Determinants

Question

Determinants cube root of unity question:
Evaluate:
| 1 w3 w5 |
|w3 1 w4 |
|w5 w5 1 |
, where w is an imaginary cube root of unity
(I know the answer is 0 but how do you solve this determinant?)

Priyanka Kedia · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

1w3w5w31w4w5w51=1w3w3w2w31w3ww3w2w3w21                   =111×w2111×w1×w21×w21                 Since w is an imaginary cube root of unity, so w3=1                   =11w211ww2w21                   =11-w3-11-w3+w2w2-w2                   =1-w3-1+w3+0                   =0⇒1w3w5w31w4w5w51=0

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Determinants cube root of unity question: Evaluate: | 1 w3 w5 | |w3 1 w4 | |w5 w5 1 | , where w is an imaginary cube root of unity. (I know the answer is 0 but how do you solve this determinant?)

Determinants cube root of unity question: Evaluate:
| 1 w³ w⁵ |
|w³1 w⁴ |
|w⁵ w⁵ 1 |

, where w is an imaginary cube root of unity.
(I know the answer is 0 but how do you solve this determinant?)