NCERT Class 12 Maths · Chapter 3 Matrices · Exercise 3.2 Question 13

✅ Step-by-Step Solution

1. Multiply row by column. The upper 2×2 block is \(\left[\begin{matrix}- \sin{\left(x \right)} \sin{\left(y \right)} + \cos{\left(x \right)} \cos{\left(y \right)} & - \sin{\left(x \right)} \cos{\left(y \right)} - \sin{\left(y \right)} \cos{\left(x \right)}\\\sin{\left(x \right)} \cos{\left(y \right)} + \sin{\left(y \right)} \cos{\left(x \right)} & - \sin{\left(x \right)} \sin{\left(y \right)} + \cos{\left(x \right)} \cos{\left(y \right)}\end{matrix}\right]\).
2. Use cos(x+y)=cos x cos y−sin x sin y and sin(x+y)=sin x cos y+cos x sin y.
3. The last row and column remain (0,0,1). Thus the whole product is \(\left[\begin{matrix}\cos{\left(x + y \right)} & - \sin{\left(x + y \right)} & 0\\\sin{\left(x + y \right)} & \cos{\left(x + y \right)} & 0\\0 & 0 & 1\end{matrix}\right]\)=F(x+y).