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Matrices
NCERT Class 12
CBSE
Grade 12

Question:

Using Principle of Mathematical Induction, prove that if $A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$, then $A^n = \begin{bmatrix} \cos n\theta & \sin n\theta \\ -\sin n\theta & \cos n\theta \end{bmatrix}$ for all $n \in \mathbb{N}$.

Step-by-Step Solution

Base case $n=1$ holds. Assume $P(k)$ true: $A^k = \begin{bmatrix} \cos k\theta & \sin k\theta \\ -\sin k\theta & \cos k\theta \end{bmatrix}$. [1.0 Mark]
$A^{k+1} = A^k A = \begin{bmatrix} \cos k\theta \cos\theta - \sin k\theta \sin\theta & \cos k\theta \sin\theta + \sin k\theta \cos\theta \\ -(\sin k\theta \cos\theta + \cos k\theta \sin\theta) & \cos k\theta \cos\theta - \sin k\theta \sin\theta \end{bmatrix}$. [1.5 Marks]
$A^{k+1} = \begin{bmatrix} \cos(k+1)\theta & \sin(k+1)\theta \\ -\sin(k+1)\theta & \cos(k+1)\theta \end{bmatrix}$. Proved by PMI! [0.5 Mark]

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🎯 Official CBSE Marking Scheme:
Stating base case & inductive hypothesis: 1.0 Mark
Evaluating $A^k A$ using compound angle formulas: 1.5 Marks
Concluding PMI proof: 0.5 Mark

Correct Answer:
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