The value of the determinant A=| 3 3 3 2 ω ω^2 1 ω^2 ω | where,

The value of the determinant A=| 3 3 3 2 ...

The value of the determinant $\mathrm{A}=\left|\begin{array}{ccc}3 & 3 & 3 \\ 2 & \omega & \omega^{2} \\ 1 & \omega^{2} & \omega\end{array}\right|$ where, $\omega$ is a complex cube root of unity is
(a) 0
(b) 3
(c) $\frac{-1+\sqrt{3} \mathrm{i}}{2}$
(d) $\pm 12 \mathrm{i} \sqrt{3}$

Key Concepts

Complex Cube Roots of Unity

Complex cube roots of unity are the solutions to the equation x³ = 1 in the complex plane. Besides the trivial solution x = 1, the other two solutions are non-real and satisfy special properties, such as summing to -1 when added to the real root, and particularly satisfying the relation 1 + ? + ?² = 0. This relation is often used to simplify expressions and computations in problems involving symmetric patterns or cyclic structures within matrices.

Determinant

A determinant is a scalar value computed from a square matrix, which provides important information such as whether the matrix is invertible, and how the matrix scales volumes in linear transformations. Evaluating a determinant involves combining the entries of the matrix in specific ways, with operations like expansion by minors or using row operations to simplify the computation.

Transcript

00:01 So in the given question we have a determinant, we have a determinant a which is given as 3, 3, 3, 2 omega omega square, 1 omega square omega and we are told that we are told to find the value of this determinant if it is given that omega is the complex cube root of 1.

00:32 Complex cube root of 1.

00:38 So what we can do over here is we can simplify this matrix first, right? so what we are going to do is to do a row operation on this determinant.

00:55 So what we would have is first we can see that we can take three as a common factor from the first row.

01:04 So 1 -1 -1 -2 omega -o omega square, 1 omega square omega and after taking 3 as a common factor, let's write, let's change the first row to r1 plus r2 plus r3.

01:23 Right? so what we would have then is 1 plus 2 plus 1 which is 4 and 1 plus omega plus omega square and 1 plus omega plus omega square.

01:36 So now we should know a property of the complex cube root of omega which is that when we take the sum of 1 omega and omega square what we get is the number 0.

01:51 So the value of this is the number 0.

01:51 So the value of this is the is 1 plus omega plus omega is equal to 0 and this is a property of the complex cube root of 1 right so then we can write this determinant as 4 0 0 2 omega omega square 1 omega square omega right so what we can do first and what we can do next is we can find the determinant easily now so we have a three outside the determinant right we have a three over here so it is three times the determinant is four times omega square minus omega raised to four so this is the and this is equal to 12 times omega square minus omega raised to 4...

Please give Ace some feedback