GATE 2023 | MATHS | Practice Problem for DA Paper

Question

Let \(M = \begin{bmatrix} 4 & -3 \\ 1 & 0 \end{bmatrix}\). Consider the following statements:

\(P:\ M^8 + M^{12}\) is diagonalizable.

\(Q:\ M^7 + M^9\) is diagonalizable.

Which of the following statements is correct?

(A) $𝑃$ is $TRUE$ and $𝑄$ is $FALSE$

(B) $𝑃$ is $FALSE$ and $𝑄$ is $TRUE$

(C) Both $𝑃$ and $𝑄$ are $FALSE$

(D) Both $𝑃$ and $𝑄$ are $TRUE$