If `A=[(1,2),(3,4)],B=[(a, 0),(0,b)]` where `a,b, in N` If AB=BA then three exists

Options

A. only one B
B. infinitely many B's
C. more than one but infinite B's
D. not N exists

Answer

To solve the problem, we need to find the condition under which the matrices \( A \) and \( B \) commute, i.e., \( AB = BA \).

Given:
\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \]
where \( a, b \in \mathbb{N} \).

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

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 each element:
- The element at (1,1) is \( 1 \cdot a + 2 \cdot 0 = a \).
- The element at (1,2) is \( 1 \cdot 0 + 2 \cdot b = 2b \).
- The element at (2,1) is \( 3 \cdot a + 4 \cdot 0 = 3a \).
- The element at (2,2) is \( 3 \cdot 0 + 4 \cdot b = 4b \).

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

### Step 2: Calculate \( BA \)

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

Calculating each element:
- The element at (1,1) is \( a \cdot 1 + 0 \cdot 3 = a \).
- The element at (1,2) is \( a \cdot 2 + 0 \cdot 4 = 2a \).
- The element at (2,1) is \( 0 \cdot 1 + b \cdot 3 = 3b \).
- The element at (2,2) is \( 0 \cdot 2 + b \cdot 4 = 4b \).

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

### Step 3: Set \( AB = BA \)

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

This gives us the following equations:
1. \( a = a \) (always true)
2. \( 2b = 2a \)
3. \( 3a = 3b \)
4. \( 4b = 4b \) (always true)

From the second equation, we can simplify:
\[ b = a \]

From the third equation, we also find:
\[ a = b \]

### Conclusion

Thus, the condition for \( AB = BA \) is that \( a = b \). Since both \( a \) and \( b \) are natural numbers, there exists an infinite number of pairs \( (a, b) \) such that \( a = b \).