Subordinate matrix norm properties proof

I have to show that the subordinate matrix norm is indeed a norm so I have to show that it satisfies the 3 properties.

I've shown how it satisfies 2 of these properties but I'm stuck with this property:
||A|| > 0 unless A is the zero matrix.

How do I show that this is true?

Quote:

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Originally Posted by CourtneyMoon

||A|| > 0 unless A is the zero matrix. How do I show that this is true?

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Hint If there exists , so where is the l-th vector of the canonical basis.