Please help me with the following problem
Given $m$ $n\times n$ matrices $A_1,A_2,...,A_m$, where $m>2^n$
Assuming that $A_i^2=O$ for all $i=1,2...,n$.
Prove that $A_1A_2...A_n=O$
Linear Algebra: Prove matrices equality
3
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linear-algebra
matrices
matrix-rank
1 Answers
3
This is not true. Let's take $n = 2$. Set
$$ A_1 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, A_2 = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}. $$
Then $A_1^2 = A_2^2 = 0$ and $A_1 A_2 A_1 = A_1$ so in particular we can form the product $A_1 A_2 A_1 A_2 A_1 = A_1$ which is non-zero.