Let $A$ be a real $4\times 4$ matrix with characteristic polynomial $(x^2+1)^2$. Which of the following is true? Let $A \in \mathbb R^{4\times 4}$ have characteristic polynomial $(x^2+1)^2$. Which of the following is true?


*

*$A$ is diagonalizable over $\mathbb C$ but not over $\mathbb R$

*$A$ is nilpotent

*$A$ is invertible 

*There is no such matrix $A$.
 A: $A$ is invertible for sure.

Given, characteristic equation is $(x^2+1)^2=0$.
$\implies (x^2+1)^2=0$
$\implies (x^2+1)=0$
$\implies x= i, i, -i, -i$
Therefore determinant of $A$ is: 
$\implies \det(A)=(i)^2\times (-i)^2=-1\times-1=1$
Since, matrix $A$ is non-singular, so, inverse of matrix $(i.e., A^{-1})$ is always exists.
A: $$P_A(x)=(x^2+1)^2=x^4+2x^2+\color{red}1$$
$$\operatorname{det}(A)=1$$
$A$ is invertible.
A: $(x^2+1)^2 = 0 \implies x^4+2x^2+1 = 0$
By Cayley–Hamilton theorem, $A^4 + 2 A^2 +I = 0$ , where $I$ is unit matrix and $0$ is zero matrix.
then $A \times (A^3 +2A) = -I \implies A \times (-A^3 -2A) = I$
So $A$ is invertible.
A: *

*If $A$ has same minimal polynomial and characteristic polynomial then
$A$ can not be diagonalizable.


Since we know that $A$ is diagonalizable iff minimal polynomial is product
of distinct linear factor.



*

*$A$ can not be nilpotent since eigen value of a nilpotent matrix are $0$
only





*

*$A$ is invertible for sure.



Proof of invertibility:
Given, characteristic equation is $(x^2+1)^2=0$.
$\implies (x^2+1)^2=0$
$\implies (x^2+1)=0$
$\implies x= i, i, -i, -i$
Therefore determinant of $A$ is:
$\implies \det(A)=(i)^2\times (-i)^2=-1\times-1=1\neq0$
And we know that $A$ is invertible iff $\det A\neq0$
Therefore matrix $A$ is  invertible.



*

*Such a matrix always exists


For Example
$$ A=\begin{bmatrix}
0&0&0&-1\\1&0&0&0\\0&1&0&-2\\0&0&1&0\end{bmatrix}$$
Then Characteristics polynomial of $A$ is $(x^2+1)^2$.
A: I don't see anyone having addressed the nilpotency. Can be nilpotent only if at least one eigenvalue is 0, but if $k$ eigenvalues were 0 we could factor out $x^k$ from the characteristic polynomial but we can't get even one out.
