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The product of a p x q and a r x s matrix is defined only if q = r and, if so, it is a p x s matrix.
That is because that is how the product and quotient are defined.
That's the way it is defined.
It is equal to 1, but not always: the reciprocal must be defined for the equality and the reciprocal is not always defined.
If by "triple dot product" you mean u·v·w, then no, because that would imply the existence of a dot product between a vector and a scalar.