Rank of a Matrix under certain conditions

Question

I am a little confused about the rank of a matrix. When does the rank of a matrix equals to zero? Is rank of a matrix equal to zero when it is a zero matrix or the matrix has no elements in it? Thank you.

Answer 1

The rank of a matrix is the largest amount of linearly independent rows or columns in the matrix. So if a matrix has no entries (i.e. the zero matrix) it has no linearly lindependant rows or columns, and thus has rank zero. If the matrix has even just $1$ entry, then we have a linearly independent row and column, and the rank is thus $1$, so in conclusion, the only rank $0$ matrix is the zero matrix.

Answer 2

We have a theorem stating $Rank(M) + Nul(M) = dim(M)$. So if the basis for the null space is the standard basis (and if the matrix isn't square, an appropriate subset of the standard basis), we have $Nul(M) = dim(M)$. So $Rank(M) = 0$.

Remember that the null space is defined as $Ker(M) = \{ v: Mv = 0 \}$, for $v$ in the vector space.