Consider a matrix $P$ whose only eigenvectors are the multiples of $\begin{bmatrix} 1 \\ 4 \end{bmatrix}$. Consider the following statements. (I) $P$ does not have an inverse (II) $P$ has a repeated eigenvalue (III) $P$ cannot be diagonalized Which one of the following options is correct?

The matrix $P$ has eigenvectors that are only scalar multiples of $\begin{bmatrix} 1 \\ 4 \end{bmatrix}$, which means $P$ is a $2 \times 2$ matrix with only one linearly independent eigenvector (geometric multiplicity = 1).

• Statement (I): $P$ does not have an inverse. This is not necessarily true. A matrix is invertible if all its eigenvalues are non-zero. For example, $P = \begin{bmatrix} 1 & 0 \\ 4 & 1 \end{bmatrix}$ has eigenvalues $\lambda = 1, 1$ (both non-zero) and only one linearly independent eigenvector $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$, and its determinant is $1 \neq 0$ (invertible). Hence, Statement (I) is not necessarily true.

• Statement (II): $P$ has a repeated eigenvalue. If $P$ had two distinct eigenvalues, it would have two linearly independent eigenvectors. Since there is only one linearly independent eigenvector, $P$ must have a repeated eigenvalue. Thus, Statement (II) is necessarily true.

• Statement (III): $P$ cannot be diagonalized. A $2 \times 2$ matrix is diagonalizable if and only if it has 2 linearly independent eigenvectors. Since $P$ has only 1 linearly independent eigenvector, it cannot be diagonalized. Thus, Statement (III) is necessarily true.

Therefore, Only II and III are necessarily true, which corresponds to option D.