Question

Let $A_{m \times n}$ be any matrix and Rank ($A$) = 0, then matrix $A$ is}

• A. Invertible
• B. Identity matrix
• C. Non-singular
• D. Zero-matrix

Hint:

Remember the rank inequality:
For any non-zero matrix \( A \neq \mathbf{0} \):
\[ \text{Rank}(A) \ge 1 \]
Thus, \( \text{Rank}(A) = 0 \iff A = \mathbf{0} \) (the zero-matrix).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The rank of a matrix $A$ (denoted as \( \text{Rank}(A) \)) is a fundamental algebraic property defined as the maximum number of linearly independent row vectors or column vectors in the matrix.

Step 2: Detailed Explanation:
Let us analyze the properties of a matrix with rank 0:
- By definition, the rank of a matrix is the dimension of the vector space spanned by its rows or columns (the column space).
- Let \( A_{m \times n} \) be a matrix. If \( \text{Rank}(A) = k \), it means there exists at least one linearly independent subset of $k$ rows or columns.
- A single vector is linearly dependent if and only if it is the zero vector.
- If \( \text{Rank}(A) = 0 \), then the dimension of the column space must be 0.
The only vector space with a dimension of 0 is the trivial vector space containing only the zero vector, \( \{ \mathbf{0} \} \).
- This implies that every single row and column vector in $A$ must be a zero vector.
- Therefore, all entries in the matrix must be zero:
\[ a_{ij} = 0 \quad \text{for all } i, j \]
This describes the Zero-matrix.
- If a matrix has even one non-zero entry, its rank must be at least 1. Thus, the only matrix with a rank of 0 is the zero-matrix.

Step 3: Final Answer:
Matrix $A$ is a Zero-matrix.