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

Explanation

To solve the problem, we need to analyze the matrices \( A \) and \( B \) and find the conditions under which the products \( AB \) and \( BA \) are equal.

Given:
\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \]

### Step 1: Calculate \( AB \)

To find \( AB \):
\[ AB = A \cdot B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \cdot \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \]

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 2: Calculate \( BA \)

Now, we calculate \( BA \):
\[ BA = B \cdot A = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \cdot \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \]

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 3: Set \( AB = BA \)

Now we set the two products equal to each other:
\[ \begin{pmatrix} a & 2b \\ 3a & 4b \end{pmatrix} = \begin{pmatrix} a & 2a \\ 3b & 4b \end{pmatrix} \]

From this, we can equate the corresponding elements:
1. \( a = a \) (always true)
2. \( 2b = 2a \)
3. \( 3a = 3b \)
4. \( 4b = 4b \) (always true)

### Step 4: Solve the equations

From \( 2b = 2a \):
\[ b = a \]

From \( 3a = 3b \):
\[ a = b \]

Thus, we conclude that \( a = b \).

### Conclusion

Since \( a \) and \( b \) are natural numbers, the condition \( a = b \) holds true. Therefore, the correct option is:

**Option C: There exists infinitely many \( b \) such that \( AB = BA \)**, since for any natural number \( a \), \( b \) can be equal to \( a \).