If `f (theta) = [[cos^(2) theta , cos theta sin theta,-sin theta],[cos theta sin theta , sin^(2) theta , cos theta ],[sin theta ,-cos theta , 0]] ` "then " f (pi/7) ` is

Options

A. symmetric
B. skew-symmetric
C. singular
D. non-singular

Explanation

To solve the problem, we need to evaluate the matrix \( f(\theta) \) at \( \theta = \frac{\pi}{7} \). The matrix is defined as:

\[ f(\theta) = \begin{bmatrix} \cos^2 \theta & \cos \theta \sin \theta & -\sin \theta \\ \cos \theta \sin \theta & \sin^2 \theta & \cos \theta \\ \sin \theta & -\cos \theta & 0 \end{bmatrix} \]

### Step 1: Substitute \( \theta = \frac{\pi}{7} \)

We will substitute \( \theta \) with \( \frac{\pi}{7} \) in the matrix.

\[ f\left(\frac{\pi}{7}\right) = \begin{bmatrix} \cos^2\left(\frac{\pi}{7}\right) & \cos\left(\frac{\pi}{7}\right) \sin\left(\frac{\pi}{7}\right) & -\sin\left(\frac{\pi}{7}\right) \\ \cos\left(\frac{\pi}{7}\right) \sin\left(\frac{\pi}{7}\right) & \sin^2\left(\frac{\pi}{7}\right) & \cos\left(\frac{\pi}{7}\right) \\ \sin\left(\frac{\pi}{7}\right) & -\cos\left(\frac{\pi}{7}\right) & 0 \end{bmatrix} \]

### Step 2: Calculate the elements of the matrix

Now we need to compute the values of \( \cos\left(\frac{\pi}{7}\right) \) and \( \sin\left(\frac{\pi}{7}\right) \). For the sake of this solution, we will denote:

- \( c = \cos\left(\frac{\pi}{7}\right) \)
- \( s = \sin\left(\frac{\pi}{7}\right) \)

Thus, we can rewrite the matrix as:

\[ f\left(\frac{\pi}{7}\right) = \begin{bmatrix} c^2 & cs & -s \\ cs & s^2 & c \\ s & -c & 0 \end{bmatrix} \]

### Step 3: Check properties of the matrix

To determine if the matrix is symmetric, skew-symmetric, singular, or non-singular, we will compute the determinant of the matrix.

### Step 4: Calculate the determinant

Using the determinant formula for a \( 3 \times 3 \) matrix:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix:

\[ \text{det}(f) = c^2 \left( s^2 \cdot 0 - c \cdot (-c) \right) - cs \left( cs \cdot 0 - s \cdot (-s) \right) + (-s) \left( cs \cdot (-c) - s^2 \cdot c \right) \]

Calculating each term:

1. The first term simplifies to \( c^2(c^2) = c^4 \).
2. The second term simplifies to \( cs(s^2) = cs^3 \).
3. The third term simplifies to \( -s(-c^2s - s^2c) = s(c^2s + s^2c) = sc^2s + s^3c \).

Putting it all together:

\[ \text{det}(f) = c^4 - cs^3 + sc^2s + s^3c = c^4 + s^3c \]

### Step 5: Evaluate the determinant at \( \theta = \frac{\pi}{7} \)

Since we know that \( c^2 + s^2 = 1 \), we can conclude that \( \text{det}(f) \) is not equal to zero, indicating that the matrix is non-singular.

### Final Result

Thus, we conclude that:

\[ f\left(\frac{\pi}{7}\right) \text{ is a non-singular matrix.} \]