Find all possibilities for tr(A) and det(A) if $A^2=4I$.

Question

Let $A$ be an $n \times n$ matrix such that $A^2 = 4I$.

(a). Find all possible values of $tr(A)$ and $det(A)$.

(b). Show that $A$ is diagonalizable.

Note: As $A^2 = 4I$, it gives that $A$ is non-singular with $\lambda \in \lbrace\pm 2 \rbrace$. So, $tr(A) \in \lbrace -2n,-2n+2,\cdots,2n-2,2n\rbrace$ and $det(A)\in \lbrace -2^n,2^n \rbrace$. Is it true? I have no ideas for the second part.

Hint for (b) :minimal polynomial of A divides $x^2-4$ and the latter has simple roots! — Chinnapparaj R, Oct 15 '23 at 14:34
@ChinnapparajR Thanks, but I don't want to use minimal polynomial. — 6-0, Oct 15 '23 at 14:43
note that $B:=\frac{1}{2}A$ is an involution, which is always diagonalizable over a field not of characteristic 2, since $\frac{1}{2}\big(B+I)$ is idempotent. — user8675309, Oct 15 '23 at 15:44

score 0 · Answer 1 · answered Oct 15 '23 at 22:34

0

Over the complex numbers a matrix is diagonalizable if and only if its minimal polynomial is has roots of multiplicity 1 (see here for details).

In your case the minimal polynomial divides $x^2-4$ and so its roots are all of multiplicity 1.

answered Oct 15 '23 at 22:34

Sam Ballas

768

Find all possibilities for tr(A) and det(A) if $A^2=4I$.

1 Answers1