Find the dimension of the vector space of 7×7 matrices with zero trace

Question

This was on a practice problem in a textbook I decided to take a look at and haven't been able to solve it. Any help?

Answer 1

$V=M_{7\times 7}=\{A_{7\times 7}=[a_{ij}]~|~\sum\limits_{i=1}^{7}a_{ii}=0, ~a_{ij}\in\mathbb{R}\}$ over the field $\mathbb{R}$.

dim($V$)$=49-1=48$, as $a_{11}=-\sum\limits_{i=2}^{7}a_{ii}$.

Observe that $\{E_{ij}, F_{ii}\}$ forms basis for $V$, where $E_{ij}$ is a matrix in which $1$ is placed at $(i,j)^{th}$ position, $i\neq j$ and rest all are zeros, and $F_{ii}$ is a matrix in which $-1$ is placed at $(1,1)$ position and $1$ is placed at $(i,i)$ position and rest all are zeros.