Rank Calculation for Null Matrix

A null matrix, often denoted as $O$ or $A$ in this context, is a matrix where all its entries are zero.

The rank of a matrix is defined as the maximum number of linearly independent row vectors (or column vectors) in the matrix. It also corresponds to the dimension of the row space or column space.

Determining Rank of Zero Matrix

• For a null matrix $A$, every row is a zero vector. For example, if $A$ is $3 \times 3$, it looks like: $ A = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} $
• The zero vector is considered linearly dependent. Since all rows (or columns) are zero vectors, there are no linearly independent rows (or columns) that form a basis for a non-zero dimensional vector space.
• The dimension of the vector space spanned by the zero vector(s) is 0.
• Therefore, the rank of any null matrix $A$ is 0.

This matches Option 2.