NCERT Class 12 Maths · Chapter 3 Matrices · Exercise 3.3 Question 6

✅ Step-by-Step Solution

1. Part (1): Aᵀ=\(\left[\begin{matrix}\cos{\left(\alpha \right)} & - \sin{\left(\alpha \right)}\\\sin{\left(\alpha \right)} & \cos{\left(\alpha \right)}\end{matrix}\right]\).
2. Multiply row by column. Diagonal entries are sin²α+cos²α=1; off-diagonal entries cancel, giving AᵀA=\(\left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]\).
3. Part (2): Aᵀ=\(\left[\begin{matrix}\sin{\left(\alpha \right)} & - \cos{\left(\alpha \right)}\\\cos{\left(\alpha \right)} & \sin{\left(\alpha \right)}\end{matrix}\right]\).
4. Multiply row by column. Diagonal entries are sin²α+cos²α=1; off-diagonal entries cancel, giving AᵀA=\(\left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]\).