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I know what are triangular matrices, hence I asked if those functions should return bool, i.e., to return true if the current matrix is an upper (or lower) triangular matrix.

If they are supposed...

This is wrong:

#include <iostream>
#include <vector>


using namespace std;


#ifndef MATRIX_H

You cannot, but I can, yet I won't ;)

You probably just made a mistake somewhere.

Well, why did you not just write the body of the function template instead of repeating the return type, name, etc? Obviously, the reason is that if you did that, the compiler would have no way of...

You are, after all, declaring/defining a function template.

Notice that the forward declaration and the definition does not match.

If you only want certain types, perhaps you should overload instead.