Exercise 21, § 4

Let $n\geqslant 1$ be an integer.

a) For every integer $r > n^2$ and every family $(a_1, . . . , a_r)$ of elements of $\mathscr{L}(\mathbf{C}^n)$, one has

b) Let $r(n)$ denote the smallest integer $r\geqslant 1$ such that formula (4) is valid for every family $(a_i)_{1\leqslant i\leqslant r}$. One has $r(n)\geqslant r(n-1) + 2$. (Consider a family $(a_1, . . . , a_{r(n-1)-1})$ in $\mathscr{L}(\mathbf{C}^{n-1})$ such that the above expression, denoted by $b$, is nonzero; interpreting the elements of $\mathscr{L}(\mathbf{C}^n)$ as matrices, choose $i$ and $j$ such that $b_{i,j}$ is nonzero; view $a_i$ and $b$ as elements of $\mathscr{L}(\mathbf{C}^n)$ by adding a row and a column of zeros, then define matrices $a_{r(n-1)}$ and $a_{r(n-1)+1}$ in such a way that expression (4) for $(a_1, . . . , a_{r(n-1)+1})$ is nonzero.)