Question

Let $A = \begin{bmatrix} \cos^2 x & \sin^2 x \\ \sin^2 x & \cos^2 x \end{bmatrix}$ and $B = \begin{bmatrix} \sin^2 x & \cos^2 x \\ \cos^2 x & \sin^2 x \end{bmatrix}$. Then the determinant of the matrix $A+B$ is ________.

• A. 1
• B. 10
• C. 0
• D. 2

Hint:

If two rows or columns of a matrix are identical, the determinant is always zero.

Answer 1

The Correct Option is C

Step 1: Concept
Matrix addition and finding the determinant. Use the identity $\sin^2 x + \cos^2 x = 1$.
Step 2: Analysis
$A + B = \begin{bmatrix} \cos^2 x + \sin^2 x & \sin^2 x + \cos^2 x \\ \sin^2 x + \cos^2 x & \cos^2 x + \sin^2 x \end{bmatrix}$. $A + B = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$.
Step 3: Calculation
$\text{det}(A+B) = (1 \times 1) - (1 \times 1)$. $\text{det}(A+B) = 1 - 1 = 0$.
Step 4: Conclusion
Hence, the determinant is 0.
Final Answer:(C)