Question

Let X be a n-square matrix such that Y = X + 8I. Which of the following property will exist?

(a) idempotent

(b) Y transpose is nilpotent

(c) X nilpotent

(d) Y inverse

The question was posed to me during an interview.

This intriguing question originated from Permutation Groups topic in division Groups of Discrete Mathematics

Answer 1

The correct option is (b) Y transpose is nilpotent

The explanation: Suppose, we have a matrix

\(a=\begin{bmatrix}

1 & 0\\

2 & 1\\

\end{bmatrix} \)

 then Y^2 will not resulting in Y, hence it is not idempotent, Y^2 is not 0 and so it is not nilpotent. But, as Y is a square matrix, by the property inverse will exist in this case.