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

JEE Advanced · 2011 · 10 Apr 2011 · Paper 2 · Determinants

Let

ω \neq = 1

be a cube root of unity and

S

be the set of all non-singular matrices of the form

1 ω ω^{2} a 1 ω b c 1,

where each of

a

b

, and

c

is either

ω

ω^{2}

. Then the number of distinct matrices in the set

S

(A)2✓ Correct
(B)6
(C)4
(D)8

Answer: (A)

Solution

For the matrix to be non-singular, its determinant must be non-zero.

Expanding the determinant:

1 ω ω^{2} a 1 ω b c 1 = 1 + a c ω^{2} + b ω^{2} - b ω^{2} - c ω - aω

Simplifying:

1 + a c ω^{2} - (a + c) ω \neq = 0

.

Each of

a, b, c

can be

ω

ω^{2}

, giving

2^{3} = 8

total combinations.

Testing which combinations make the determinant zero, we find that 6 combinations result in a singular matrix.

Therefore, only 2 combinations yield non-singular matrices:

(a, b, c) = (ω, ω^{2}, ω)

and

(ω, ω, ω)

JEE Advanced · 2011 · 10 Apr 2011 · Paper 2 · Single correct option

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JEE Advanced · 2011 · 10 Apr 2011 · Paper 2 · Determinants

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