Let A = ( a i j ) {\displaystyle {}A={\left(a_{ij}\right)}} and B = ( b i j ) {\displaystyle {}B={\left(b_{ij}\right)}} be square matrices of length n {\displaystyle {}n} . Suppose that a i j = 0 {\displaystyle {}a_{ij}=0} holds for j ≤ i + d {\displaystyle {}j\leq i+d} , and b i j = 0 {\displaystyle {}b_{ij}=0} holds for j ≤ i + e {\displaystyle {}j\leq i+e} for some d , e ∈ Z {\displaystyle {}d,e\in \mathbb {Z} } . Show that the entries c i j {\displaystyle {}c_{ij}} of the product A B {\displaystyle {}AB} fulfill the condition c i j = 0 {\displaystyle {}c_{ij}=0} for j ≤ i + d + e + 1 {\displaystyle {}j\leq i+d+e+1} .
