How to show that $\operatorname{rank}(A) \le 4$ and $A + A^T$ is irreversible

Question

$A \in \mathbb{R}^{10,10}$, vectors $x_1,x_2, \ldots x_7 \in \mathbb{R}^{10}$ are linearly independent, $Ax_1 = Ax_2 = \ldots = Ax_7$. I need to show that

a) $\operatorname{rank}(A) \le 4$

b) $A + A^T$ is irreversible

I don't know even how to begin. Can you help me?

Answer 1

Since $Av_1\dots Av_7$ are equal, it follows that the dimension of the range of $A$ is less than $4$. Hence, the rank of $A$ and of $A^T$ is less than $4$. The rank of $A+A^T$ is less than $8$, and $A+A^T$ cannot be invertible.

To see that rank of $A$ can be $4$, consider the matrix $$ A=\pmatrix{ 1&1&1&1&1&1&1&0&0&0\\ 0&0&0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&0&0&1\\ &&&&\dots \\ }.$$