Define the function f:M3(Z)→Zf: M_3(\mathbb Z)\to \mathbb Zf:M3(Z)→Z in the following way:
f([aij]3×3)=a11+a22+a33f([a_{ij}]_{3\times 3})=a_{11}+a_{22}+a_{33}f([aij]3×3)=a11+a22+a33, where aij∈Z.a_{ij}\in\mathbb Z.aij∈Z.
This function is non-zero because f(E)=1+1+1=3≠0f(E)=1+1+1=3\ne 0f(E)=1+1+1=3=0 for the identity matrix E∈M3(Z)E\in M_3(\mathbb Z)E∈M3(Z).
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