Let $A=\begin{bmatrix}\alpha & \beta\\0 & \alpha\end{bmatrix}$ be the $n^{th}$ root of $I_2$, then choose the correct statement

Question

Let $A=\begin{bmatrix}\alpha & \beta\\0 & \alpha\end{bmatrix}$ be the $n^{th}$ root of $I_2$, then choose the correct statement (more than one correct)

-> A) if $n$ is odd, $\alpha=1,\beta=0$

-> B) if $n$ is odd, $\alpha=-1, \beta=0$

-> C) if $n$ is even, $\alpha=1, \beta=0$

-> D) if $n$ is even, $\alpha=-1, \beta=0$

My Attempt:

Taking $n=2,$ I get $\beta=0,\alpha=\pm1$

Taking $n=3,$ I get $\beta=0,\alpha=1$

Taking $n=4,$ I get $\beta=0, \alpha=\pm1$

So, I think the answer should be a), c), d)

But the answer given is a), c)

Why is d) incorrect?

Answer 1

It's a consequence of definition, more or less. The identity matrix is the multiplicative identity for matrices just as 1 is for the integers. It's really nice for an operation done on the multiplicative identity of a group to give back itself, essentially, so typically only the principal branch of even roots is considered for both 1 and the identity matrix. Having alpha equal to -1 however is arithmetically correct even if not typically considered, so I believe this is just a poorly written question.