For x a real number, the determinant of the matrix

$\begin{pmatrix} \cos x & 0 & \sin x \\0 & 1 & 0 \\-\sin x & 0 & \cos x \end{pmatrix}$ equals 

a. +1  

b. −1  

c. sin(2x)  

d. cos2x  

e. None of the provided

To determine the determinant of the given matrix 

\[ A = \begin{pmatrix} \cos x & 0 & \sin x \\ 0 & 1 & 0 \\ -\sin x & 0 & \cos x \end{pmatrix}, \]

we can proceed as follows:

1. The matrix $ A $ is a block diagonal matrix with one block being $\begin{pmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{pmatrix}$ and the other block being $1$.

2. For block diagonal matrices, the determinant of the matrix is the product of the determinants of the blocks.

First, consider the $2 \times 2$ submatrix:

\[ \begin{pmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{pmatrix}. \]

The determinant of this $2 \times 2$ matrix is calculated as:

\[ \det \begin{pmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{pmatrix} = (\cos x)(\cos x) - (\sin x)(-\sin x) = \cos^2 x + \sin^2 x. \]

Using the Pythagorean identity $\cos^2 x + \sin^2 x = 1$, we find:

\[ \det \begin{pmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{pmatrix} = 1. \]

The other block is simply $1$, and its determinant is $1$.

Therefore, the determinant of the entire matrix $ A $ is:

\[ \det(A) = \det \begin{pmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{pmatrix} \cdot 1 = 1 \cdot 1 = 1. \]

Hence, the determinant of the given matrix is $\boxed{1}$.

So, the correct answer is:

a. +1

Would you like more details or have any questions?

Here are some related questions you can ask:

1. How do you find the determinant of a $3 \times 3$ matrix?
2. What is a block diagonal matrix?
3. How do you verify the Pythagorean identity $\cos^2 x + \sin^2 x = 1$?
4. How do you compute the determinant of a matrix using cofactors?
5. What are the properties of determinants for orthogonal matrices?

**Tip:** For a block diagonal matrix, the determinant is the product of the determinants of the individual blocks. This can simplify the computation of determinants for larger matrices.

Explore how to find the determinant of a matrix $ \begin{pmatrix} \cos x & 0 & \sin x \\ 0 & 1 & 0 \\ -\sin x & 0 & \cos x \end{pmatrix} $. This problem involves trigonometric functions and matrix determinants, using the Pythagorean identity to simplify calculations. Suitable for advanced high school students and beyond.

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