Let `theta=(pi)/(5),X=[{:(,cos theta,-sin theta),(,sin theta,cos theta):}]`, O is null matrix and I is an identity of order `2 xx 2`, and if `I+X+X^(2)+....+X^(n)=0` then n can be

Options

A. 9
B. 19
C. 4
D. 29

Answer

To solve the problem, we need to find the value of \( n \) such that the equation

\[ I + X + X^2 + \ldots + X^n = 0 \]

holds true, where \( X \) is defined as:

\[ X = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \]

and \( \theta = \frac{\pi}{5} \).

### Step 1: Calculate \( X^2 \)

To find \( X^2 \), we multiply \( X \) by itself:

\[ X^2 = X \cdot X = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \cdot \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \]

Calculating the elements:

- First row, first column:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]

- First row, second column:
\[ -\cos \theta \sin \theta + \sin \theta \cos \theta = 0 \]

- Second row, first column:
\[ \sin \theta \cos \theta + \cos \theta \sin \theta = 0 \]

- Second row, second column:
\[ -\sin^2 \theta + \cos^2 \theta = \cos 2\theta \]

Thus, we have:

\[ X^2 = \begin{pmatrix} 1 & 0 \\ 0 & \cos 2\theta \end{pmatrix} \]

### Step 2: General Form of \( X^n \)

Using the properties of rotation matrices, we can deduce that:

\[ X^n = \begin{pmatrix} \cos(n\theta) & -\sin(n\theta) \\ \sin(n\theta) & \cos(n\theta) \end{pmatrix} \]

### Step 3: Substitute into the Equation

Now substituting into the equation \( I + X + X^2 + \ldots + X^n = 0 \):

\[ I + X + X^2 + \ldots + X^n = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} + \begin{pmatrix} \cos 2\theta & -\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{pmatrix} + \ldots + \begin{pmatrix} \cos n\theta & -\sin n\theta \\ \sin n\theta & \cos n\theta \end{pmatrix} = 0 \]

### Step 4: Analyze the Components

The equation can be separated into two parts (the components of the identity matrix and the cosine/sine components):

1. For the first component (identity matrix):
\[ 1 + \cos \theta + \cos 2\theta + \ldots + \cos n\theta = 0 \]

2. For the second component (sine):
\[ -\sin \theta - \sin 2\theta - \ldots - \sin n\theta = 0 \]

### Step 5: Use the Sum of Cosines Formula

The sum of cosines can be expressed using the formula for the sum of a geometric series:

\[ \sum_{k=0}^{n} \cos(k\theta) = \frac{\sin\left(\frac{(n+1)\theta}{2}\right) \cos\left(\frac{n\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \]

Setting this equal to \(-1\) (from the first component) leads us to find \( n \).

### Step 6: Substitute \( \theta = \frac{\pi}{5} \)

Substituting \( \theta = \frac{\pi}{5} \):

\[ 1 + \cos\left(\frac{\pi}{5}\right) + \cos\left(\frac{2\pi}{5}\right) + \ldots + \cos\left(\frac{n\pi}{5}\right) = 0 \]

### Step 7: Find Values of \( n \)

By evaluating the cosine terms, we find that they cancel out for certain values of \( n \). The complete cycle occurs every \( n = 9 \) (as shown in the video).

### Conclusion

Thus, the possible values of \( n \) are:

\[ n = 9, 19, 29, \ldots \]