If the matrix AB is zero then

Options

A. A=O or B=O
B. A=O and B=O
C. It is not necessary that either A=O or B=O
D. All these statements are wrong.

Answer

To solve the problem, we need to analyze the condition under which the product of two matrices \( AB \) results in a zero matrix, even if neither \( A \) nor \( B \) is a zero matrix. Here’s a step-by-step solution:

### Step 1: Understanding the Zero Matrix
The zero matrix is defined as a matrix where all entries are zero. For example, a 2x2 zero matrix looks like:
\[ 0 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

### Step 2: Setting Up Matrices A and B
Let’s assume matrices \( A \) and \( B \) are not zero matrices. For instance:
\[ A = \begin{pmatrix} 0 & 0 \\ 4 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \]

### Step 3: Multiplying Matrices A and B
To find the product \( AB \), we multiply the rows of \( A \) by the columns of \( B \):
\[ AB = \begin{pmatrix} 0 & 0 \\ 4 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \]

Calculating the entries:
- The entry at (1,1) is \( 0 \times 0 + 0 \times 1 = 0 \)
- The entry at (1,2) is \( 0 \times 0 + 0 \times 0 = 0 \)
- The entry at (2,1) is \( 4 \times 0 + 0 \times 1 = 0 \)
- The entry at (2,2) is \( 4 \times 0 + 0 \times 0 = 0 \)

Thus, we find:
\[ AB = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

### Step 4: Conclusion About Matrices A and B
From the above multiplication, we see that even though neither \( A \) nor \( B \) is a zero matrix, their product \( AB \) is still a zero matrix. This leads us to conclude that it is not necessary for either \( A \) or \( B \) to be a zero matrix for \( AB \) to be a zero matrix.

### Final Answer
Therefore, the correct condition is:
- It is **not necessary** that either \( A \) or \( B \) is a zero matrix.