Problem related to positive definite , symmetric , skew symmetric matrix Pick out the true statement(s):


*

*(a) There exist $n \times n$ matrices with real entries such that $(I-(AB-BA))^n=0$.

*(b) $A $ is a symmetric and positive definite $n\times n$ matrix then $(\operatorname{tr}(A))^n\ge n^n \det(A)$.  

*(c) Let $A$ be a $5 \times 5$ skew -symmetric matrix with real entries. Then $A$ is
singular.


a) never true
Note that $AB - BA$ can never have $1$ as an eigenvalue. Hence
$$
\text{If } \lambda \text{ is an eigenvalue of } I -(AB-BA) \text { then } \lambda \ne 0$$
c) If $\lambda$ is an eigenvalue, then so is $\lambda$. Any odd dimensional matrix must have a real eigenvalue. So zero is an eigenvalue. What can you conclude?
b) is easy
