A) 10
B) zero matrix
C) symmetric matrix
D) 4
Correct Answer: B
Solution :
Idea \[\because \]Let \[A=[{{a}_{i\,j}}]\]be \[{{a}_{m\times n}}\]matrix and k be any number, then \[{{k}_{A}}={{[k{{a}_{i\,j}}]}_{m\times n}}\]and use formula such as\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{\cos n\theta }{n}=\frac{\sin \theta }{n}=0\] Given that \[A=\left[ \begin{matrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{matrix} \right]\] \[{{A}^{n}}=\left[ \begin{matrix} \cos n\theta & \sin n\theta \\ -\sin n\theta & \cos n\theta \\ \end{matrix} \right]\] \[\frac{{{A}^{n}}}{n}=\left[ \begin{matrix} \frac{\cos n\theta }{n} & \frac{\sin n\theta }{n} \\ \frac{-\sin n\theta }{n} & \frac{\cos n\theta }{n} \\ \end{matrix} \right]\] \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{A}^{n}}}{n}=\left[ \begin{matrix} \underset{x\to \infty }{\mathop{\lim }}\,\frac{\cos n\theta }{n} & \underset{x\to \infty }{\mathop{\lim }}\,\frac{\sin n\theta }{n} \\ \underset{x\to \infty }{\mathop{\lim }}\,\frac{-\sin n\theta }{n} & \underset{x\to \infty }{\mathop{\lim }}\,\frac{\cos n\theta }{n} \\ \end{matrix} \right]\] \[=\left[ \begin{matrix} 0 & 0 \\ 0 & 0 \\ \end{matrix} \right]=\]zero matrix TEST Edge Properties of scalar multiplication and addition of matrices based questions are asked. These questions are solved with the help of properties of matrices.
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