If A=[\begin{matrix}2&0&0\\ 0&3&0\\ 0&0&4\end{matrix}], then which of the following statements are correct? 1. A^{n} will <strong>ALWAYS</strong> be singular for any positive integer n. 2. A^{n} will <strong>ALWAYS</strong> be a diagonal matrix for any positive integer n. 3. A^{n} will <strong>ALWAYS</strong> be a symmetric matrix for any positive integer n.
- A. 1 and 2 only
- B. 2 and 3 only ✓
- C. 1 and 3 only
- D. 1, 2 and 3
Correct Answer: B. 2 and 3 only
Explanation
A is a diagonal matrix with det(A) = 24 \neq 0. Therefore, A^n is non-singular, so statement 1 is false. Any integer power of a diagonal matrix yields another diagonal matrix, and every diagonal matrix is inherently symmetric. So statements 2 and 3 are correct.
Related questions on Matrices & Determinants
- Consider the determinant \Delta=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{vmatrix} If $a_{13...
- If A=\begin{pmatrix}1&0&0\\ 0&\cos~\theta&\sin~\theta\\ 0&\sin~\theta&-\cos\theta\end{pmatrix}, then which of the following are correct?...
- If X is a matrix of order 3\times3, Y is a matrix of order 2\times3 and Z is a matrix of order 3\times2, then which of the follo...
- What is the value of a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}?
- What is the value of a_{21}C_{11}+a_{22}C_{12}+a_{23}C_{13}?