Let `A=[{:(1,2),(3,4):}]and B = [{:(a,0),(0,b):}]` where a, b are natural numbers, then which one of the following is correct ?

Options

A. there cannot exist any B such that `AB = BA`
B. There exist more than one but finite number of `B`' s such that
`AB = BA`
C. there exists exactly one B such that `AB = BA`
D. there exist infinitely among `B'` s such that `AB = BA`

Explanation

To solve the problem, we need to analyze the matrices \( A \) and \( B \) and determine the conditions under which \( AB = BA \).

### Step 1: Define the matrices
Let:
\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \]
and
\[ B = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \]
where \( a \) and \( b \) are natural numbers.

### Step 2: Calculate the product \( AB \)
To find \( AB \), we multiply matrix \( A \) by matrix \( B \):
\[ AB = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \]
Calculating the elements:
- First row, first column: \( 1 \cdot a + 2 \cdot 0 = a \)
- First row, second column: \( 1 \cdot 0 + 2 \cdot b = 2b \)
- Second row, first column: \( 3 \cdot a + 4 \cdot 0 = 3a \)
- Second row, second column: \( 3 \cdot 0 + 4 \cdot b = 4b \)

Thus,
\[ AB = \begin{pmatrix} a & 2b \\ 3a & 4b \end{pmatrix} \]

### Step 3: Calculate the product \( BA \)
Now, we calculate \( BA \):
\[ BA = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \]
Calculating the elements:
- First row, first column: \( a \cdot 1 + 0 \cdot 3 = a \)
- First row, second column: \( a \cdot 2 + 0 \cdot 4 = 2a \)
- Second row, first column: \( 0 \cdot 1 + b \cdot 3 = 3b \)
- Second row, second column: \( 0 \cdot 2 + b \cdot 4 = 4b \)

Thus,
\[ BA = \begin{pmatrix} a & 2a \\ 3b & 4b \end{pmatrix} \]

### Step 4: Set \( AB = BA \)
Now, we set the two products equal to each other:
\[ \begin{pmatrix} a & 2b \\ 3a & 4b \end{pmatrix} = \begin{pmatrix} a & 2a \\ 3b & 4b \end{pmatrix} \]

### Step 5: Compare the elements
From the equality of matrices, we compare the corresponding elements:
1. From the first row, second column: \( 2b = 2a \) implies \( b = a \).
2. From the second row, first column: \( 3a = 3b \) simplifies to \( a = b \) (which is consistent with the first equation).
3. The second row, second column \( 4b = 4b \) is always true.

### Step 6: Conclusion
Since \( b = a \), and both \( a \) and \( b \) can take any natural number value, there are infinitely many pairs \( (a, b) \) such that \( AB = BA \).

Thus, the correct answer is:
**Option 4: There exist infinitely many \( B \) such that \( AB = BA \).**