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

Q: Wait, why does the $b$ term just disappear? Did I calculate the determinant wrong?

It is a beautiful moment in linear algebra! When you expand along the first row, the cofactor of $b$ is $(\omega^2 - \omega^2)$, which is zero. It is not a mistake; it is a mathematical gift that simplifies the problem significantly.

Q: Why can't $a$ or $c$ be $\omega^2$?

Because if $a = \omega^2$, then $a\omega = \omega^3 = 1$. Then the factor $(1 - a\omega)$ becomes $(1 - 1) = 0$. If any factor is zero, the determinant is zero, making the matrix singular. We want non-singular matrices, so we must reject $\omega^2$.

Q: Does the value of $b$ really not matter?

Exactly! Since the determinant is independent of $b$, any valid value of $b$ will keep the matrix non-singular. Since $b$ can be $\omega$ or $\omega^2$, we have two choices for $b$.

Question

Let $\omega 
eq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $\begin{bmatrix} 1 & a & b \ \omega & 1 & c \ \omega^2 & \omega & 1 \end{bmatrix}$ where each of $a, b$ and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is

Options:

(A) (a) 2

(B) (b) 6

(C) (c) 4

(D) (d) 8

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with finding the number of non-singular matrices $M$ in the set $S$, where:

$$M = \begin{bmatrix} 1 & a & b \ \omega & 1 & c \ \omega^2 & \omega & 1 \end{bmatrix}$$

The variables $a, b, c$ are restricted to the set $\{\omega, \omega^2\}$, where $\omega$ is a complex cube root of unity. Recall that $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$.

### The Vanishing Act

A matrix is non-singular if and only if its determinant is non-zero. We calculate the determinant by expanding along the first row:

$$\det(M) = 1(1 - c\omega) - a(\omega - c\omega^2) + b(\omega^2 - \omega^2)$$

Observe that the coefficient of $b$ is $(\omega^2 - \omega^2) = 0$. This implies that the determinant is independent of the value of $b$.

Since $b$ can be either $\omega$ or $\omega^2$, there are $2$ possible choices for $b$ that do not affect the non-singularity of the matrix.

### The Factorization

With the $b$ term eliminated, the determinant simplifies to:

$$\det(M) = 1 - c\omega - a\omega + ac\omega^2$$

We can group the terms to factor the expression:

$$\det(M) = (1 - a\omega) - c\omega(1 - a\omega)$$

Factoring out $(1 - a\omega)$, we obtain the elegant result:

$$\det(M) = (1 - a\omega)(1 - c\omega)$$

For the matrix to be non-singular, we require $\det(M) 
eq 0$. This condition is satisfied if and only if $(1 - a\omega) 
eq 0$ and $(1 - c\omega) 
eq 0$.

### The Constraints

We evaluate the constraints on $a$ and $c$ given that $a, c \in \{\omega, \omega^2\}$.

If $a = \omega^2$, then $a\omega = \omega^3 = 1$. This results in $(1 - a\omega) = (1 - 1) = 0$, which makes the determinant zero. Therefore, $a$ must be $\omega$.

Similarly, if $c = \omega^2$, then $c\omega = \omega^3 = 1$, leading to $(1 - c\omega) = 0$. Thus, $c$ must also be $\omega$.

### The Final Count

We have determined that $a$ has $1$ valid choice ($\omega$) and $c$ has $1$ valid choice ($\omega$). As established earlier, $b$ has $2$ valid choices ($\omega$ or $\omega^2$).

The total number of non-singular matrices is the product of the number of choices for each variable:

$$	ext{Total} = 1 	imes 2 	imes 1 = 2$$

The total number of non-singular matrices is **2**.

Select Answer:

Visualized Solution

Define the Matrix and Constraints

Condition for Non-Singular Matrix

Expand the Determinant $det (M)$

Evaluate the $b$ Term

Simplify the Determinant Expression

Factorize the Determinant

Apply the Non-Singular Condition

Solve for Allowed Values of $a$

Solve for Allowed Values of $c$

Determine Choices for $b$

Calculate Total Number of Matrices

The Sigma Insight: Properties of Determinants

Analyzing the Setup

The Vanishing Act

The Factorization

The Constraints

The Final Count

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Select Answer:

Visualized Solution

Define the Matrix and Constraints

Condition for Non-Singular Matrix

Expand the Determinant .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } det(M)

Evaluate the .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b Term

Simplify the Determinant Expression

Factorize the Determinant

Apply the Non-Singular Condition

Solve for Allowed Values of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a

Solve for Allowed Values of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } c

Determine Choices for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b