Question

If \(A\) is a null Matrix, then its rank is :

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. Infinite

Hint:

A null matrix contains only zeros. Since there are no non-zero independent rows or columns: \[ \text{Rank of null matrix}=0 \]

Answer 1

The Correct Option is A

Concept: The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. A null matrix is a matrix in which every element is zero. For example: \[ \begin{bmatrix} 0 & 0 0 & 0 \end{bmatrix} \] Since every row and every column contains only zeros:
• no row is linearly independent,
• no column is linearly independent. Hence the rank becomes zero.

Step 1: Understanding a null matrix. A null matrix has all entries equal to zero. Example: \[ A= \begin{bmatrix} 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 \end{bmatrix} \] Every row vector here is: \[ (0,0,0) \] and every column vector is also: \[ (0,0,0) \]

Step 2: Understanding rank. Rank means the maximum number of linearly independent rows or columns. But in a null matrix:
• all rows are zero rows,
• all columns are zero columns,
• zero vectors are always linearly dependent. Therefore there are no independent rows or columns.

Step 3: Finding the rank. Hence: \[ \text{Rank}(A)=0 \] Therefore: \[ \boxed{0} \]

Step 4: Checking each option carefully.
• Option \(1\): \(0\) \(\rightarrow\) Correct
• Option \(2\): \(1\) \(\rightarrow\) Incorrect
• Option \(3\): \(2\) \(\rightarrow\) Incorrect
• Option \(4\): Infinite \(\rightarrow\) Incorrect Thus the correct answer is: \[ \boxed{(1)\;0} \]