If ωis a complex cube root of unity then the value of

If $\omega$ is a complex cube root of unity then the value of $\left|\begin{array}{ccc}1+\omega^{10} & 1+\omega^{11} & \omega^{10}+\omega^{11} \\ \omega^{8} & \omega^{7} & \omega^{8} \\ \omega^{13} & \omega^{14} & \omega^{14}\end{array}\right|$ is (a) $-3 \omega$ (b) 0 (c) $\omega$ (d) $-\omega^{2}$

If $\omega$ is a complex cube root of unity then the value of $\left|\begin{array}{ccc}1+\omega^{10} & 1+\omega^{11} & \omega^{10}+\omega^{11} \\ \omega^{8} & \omega^{7} & \omega^{8} \\ \omega^{13} & \omega^{14} & \omega^{14}\end{array}\right|$ is
(a) $-3 \omega$
(b) 0
(c) $\omega$
(d) $-\omega^{2}$

Key Concepts

Determinants of Matrices

The determinant is a scalar value that can be computed from the elements of a square matrix and provides important properties of the matrix, such as whether the matrix is invertible. Determinants obey various algebraic properties, including linear dependency among rows or columns leading to a zero determinant, and their calculation often involves expansion by minors or the use of row or column operations to simplify the matrix.

Cube Roots of Unity

Cube roots of unity are the solutions to the equation x³ = 1 in the complex numbers. They consist of 1 and two non-real numbers, typically denoted ? and ?², which satisfy properties such as ?³ = 1 and 1 + ? + ?² = 0. These properties are often used to simplify expressions involving higher powers of ? by reducing the exponents modulo 3.

Modular Exponentiation in Complex Roots of Unity

When dealing with powers of ?, where ? is a cube root of unity, exponents greater than 2 can be reduced modulo 3 because ?³ = 1. This means that ?^n can be simplified to one of ?^0, ?^1, or ?^2, which greatly simplifies computations in problems that involve powers of ?.

Transcript

00:01 So, in the given question we are told that omega is a complex cube root of unity.

00:08 If omega is a complex cube root of unity, then we are told to evaluate the determinant that is given as complex cube root of unity means it is the complex cube root of 1.

00:29 And if that's the case, we are told to evaluate the determinant 1 plus omega raised to 10, 1 plus omega raised to 11, omega raised to 10 plus omega raised to 11.

00:43 The second row is omega raised to 8, omega raised to 7, omega raised to 8.

00:50 The third row is omega raised to 13, omega raised to the power 14 and omega raised to 14 again.

01:01 And we are told to evaluate this determinant, right? so what we can do over here is we know that since omega is a cube root, a complex cube root of unity, omega cube is equal to 1.

01:18 Right? so, omega raised to a power that is 3 times n is always equal to 1 since the power is a multiple of 3.

01:29 And what we can do with this is we can write omega raised to 10 as omega raised to 9 n times omega, omega raised to 9 would then be equal to 1, right? so, omega raised to 10 is then equal to omega.

01:47 And.

01:48 If that's the case, omega raised to 11 would then be equal to omega raised to 9 times omega square which is equal to omega square and we can write omega raised to 8 as omega raised to 6 times omega square.

02:05 Similarly for omega raised to 7 we can write omega raised to 6 times omega and we have to do the same for omega raised to 13 and 14.

02:17 So, omega raised to 13 is equal to omega raised to 12 times omega and omega raised to 14 can be written as omega raised to 12 times omega square where omega raised to 6, omega raised to 12 and omega raised to 9 are all equal to 1, right? so the determinant would simplify to 1 plus omega, 1 plus omega square omega square plus omega, omega plus omega square, right? and in the second row we would have omega square, omega, omega, omega, square.

02:56 In the third row, we have omega, omega, omega square, omega, right? so this is what we have as the simplified determinant...

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