Question

What are the possible values for the rank of a \(4 \times 3\) matrix?

• A. \(0,1,2,3,4\)
• B. \(1,2,3,4\)
• C. \(0,1,2,3\)
• D. Only \(3\)

Hint:

Rank of \(m \times n\) matrix \( \leq \min(m,n)\)

Answer 1

The Correct Option is C

Concept: Rank of a matrix
The rank of a matrix is the maximum number of linearly independent rows or columns.
Step 1: General rule
For an \(m \times n\) matrix: \[ \text{rank} \leq \min(m,n) \]
Step 2: Apply to given matrix
Matrix size: \[ 4 \times 3 \] \[ \min(4,3) = 3 \]
Step 3: Possible values
Rank can be any integer from \(0\) up to \(3\): \[ 0,1,2,3 \]
Step 4: Interpretation

• Rank \(0\): zero matrix
• Rank \(3\): full column rank

Conclusion: \[ \text{Possible ranks} = 0,1,2,3 \]