Found problems: 7
Let $p \geq 3$ be a prime number and let $A$ be a matrix of order $p$ with complex entries. Assume that $\text{Tr}(A) = 0$ and $\det(A - I_p) \neq 0$. Prove that $A^p \neq I_p$.
Note: $\text{Tr}(A)$ is the sum of the main diagonal elements of $A$ and $I_p$ is the identity matrix of order $p$.
$A$ and $B$ are $2\times 2$ real valued matrices satisfying
$$\det A = \det B = 1,\quad \text{tr}(A)>2,\quad \text{tr}(B)>2,\quad \text{tr}(ABA^{-1}B^{-1}) = 2$$
Prove that $A$ and $B$ have a common eigenvector.
Suppose a $n\times n$ real matrix $A$ satisfies $\text{tr}(A)=2018$, $\text{rank}(A)=1$. Prove that $A^2=2018 A$.
Let \( A \) be a \( 2 \times 2 \) matrix with integer entries and \(\det A \neq 0\). If the sequence \(\operatorname{tr}(A^n)\), for \( n = 1, 2, 3, \ldots \), is bounded, show that
\[
A^{12} = I \quad \text{or} \quad (A^2 - I)^2 = O.
\]
Here, \( I \) and \( O \) denote the identity and zero matrices, respectively, and \(\operatorname{tr}\) denotes the trace of the matrix (the sum of the elements on the main diagonal).
Let $A,B \in \mathcal{M}_n(\mathbb{C})$ be two matrices such that $A+B=AB+BA$. Prove that:
a) if $n$ is odd, then $\det(AB-BA)=0$;
b) if $\text{tr}(A)\neq \text{tr}(B)$, then $\det(AB-BA)=0$.
Given positive integers $m,n$ and a $m\times n$ matrix $A$ with real entries.
(1) Show that matrices $X = I_m + AA^T$ and $Y = I_n + A^T A$ are invertible. ($I_l$ is the $l\times l$ unit matrix.)
(2) Evaluate the value of $\text{tr}(X^{-1}) - \text{tr}(Y^{-1})$.
Let $A$ be an $n\times n$ real matrix with minimal polynomial $x^n + x - 1$. Prove that the trace of $(nA^{n-1} + I)^{-1}A^{n-2}$ is zero.