NCERT Class 12 Maths · Chapter 3 Matrices · Exercise 3.2 Question 2

✅ Step-by-Step Solution

1. Add entries in matching positions; symbolic identities may then simplify each entry.
2. Part (1): \(\left[\begin{matrix}a & b\\- b & a\end{matrix}\right]\) + \(\left[\begin{matrix}a & b\\b & a\end{matrix}\right]\) = \(\left[\begin{matrix}2 a & 2 b\\0 & 2 a\end{matrix}\right]\).
3. Part (2): \(\left[\begin{matrix}a^{2} + b^{2} & b^{2} + c^{2}\\a^{2} + c^{2} & a^{2} + b^{2}\end{matrix}\right]\) + \(\left[\begin{matrix}2 a b & 2 b c\\- 2 a c & - 2 a b\end{matrix}\right]\) = \(\left[\begin{matrix}\left(a + b\right)^{2} & \left(b + c\right)^{2}\\\left(a - c\right)^{2} & \left(a - b\right)^{2}\end{matrix}\right]\).
4. Part (3): \(\left[\begin{matrix}-1 & 4 & -6\\8 & 5 & 16\\2 & 8 & 5\end{matrix}\right]\) + \(\left[\begin{matrix}12 & 7 & 6\\8 & 0 & 5\\3 & 2 & 4\end{matrix}\right]\) = \(\left[\begin{matrix}11 & 11 & 0\\16 & 5 & 21\\5 & 10 & 9\end{matrix}\right]\).
5. Part (4): \(\left[\begin{matrix}\cos^{2}{\left(x \right)} & \sin^{2}{\left(x \right)}\\\sin^{2}{\left(x \right)} & \cos^{2}{\left(x \right)}\end{matrix}\right]\) + \(\left[\begin{matrix}\sin^{2}{\left(x \right)} & \cos^{2}{\left(x \right)}\\\cos^{2}{\left(x \right)} & \sin^{2}{\left(x \right)}\end{matrix}\right]\) = \(\left[\begin{matrix}1 & 1\\1 & 1\end{matrix}\right]\).
6. In part (iv), use sin²x+cos²x=1, so every resulting entry is 1.