Question

If $A$ is a null matrix then

• A. $\text{Rank}(A) = 1$.
• B. $\text{Rank}(A) = 0$.
• C. $\text{Rank}(A) = 2$.
• D. $\text{Rank}(A) = 3$.

Hint:

$\text{Rank}(A) = 0 \iff A = O$ (Zero Matrix). For any non-zero matrix $A_{m \times n}$, $1 \leq \text{Rank}(A) \leq \min(m, n)$.

Answer 1

The Correct Option is B

Step 1: Concept
The rank of a matrix $A$ is defined as the maximum number of linearly independent row vectors (or column vectors) in $A$. Alternatively, it is the maximum order of any non-zero minor of $A$.

Step 2: Key Formulas and Approach
A null matrix (or zero matrix) $O_{m \times n}$ is a matrix in which all entries are identically zero: \[ A = \begin{pmatrix} 0 & 0 & \dots & 0 0 & 0 & \dots & 0 \vdots & \vdots & \ddots & \vdots 0 & 0 & \dots & 0 \end{pmatrix} \]

Step 3: Step-by-step Explanation

• Every row vector and column vector in a null matrix is the zero vector $\vec{0}$.

• A set consisting only of zero vectors is linearly dependent. The maximum number of linearly independent rows in a null matrix is $0$.

• Furthermore, every minor of any order ($1 \times 1, 2 \times 2, \dots$) formed from a zero matrix has a determinant equal to $0$.

• Since there exists no non-zero minor of order $1$ or greater, by definition: \[ \text{Rank}(A) = 0 \]
• Note that the null matrix is the only matrix that has a rank of $0$. Every non-zero matrix has a rank of at least $1$.

Step 4: Final Answer
The rank of a null matrix is always 0. Hence, Option (B) is correct.