If A is a square matrix of order `2xx2` and `B=[(1,2),(3, 4)]`, such that `AB=BA`, then A can be

Options

A. `[(1,4),(6, 7)]`
B. `[(1, 4),(7, 6)]`
C. `[(2,2),(2,4)]`
D. `[(3,4),(4,9)]`

Answer

To solve the problem, we need to find the possible forms of the square matrix \( A \) of order \( 2 \times 2 \) such that it commutes with the given matrix \( B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \). This means that \( AB = BA \).

Let \( A = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \).

### Step 1: Compute \( AB \)
We calculate \( AB \):
\[ AB = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} p \cdot 1 + q \cdot 3 & p \cdot 2 + q \cdot 4 \\ r \cdot 1 + s \cdot 3 & r \cdot 2 + s \cdot 4 \end{pmatrix} = \begin{pmatrix} p + 3q & 2p + 4q \\ r + 3s & 2r + 4s \end{pmatrix} \]

### Step 2: Compute \( BA \)
Next, we calculate \( BA \):
\[ BA = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} p & q \\ r & s \end{pmatrix} = \begin{pmatrix} 1 \cdot p + 2 \cdot r & 1 \cdot q + 2 \cdot s \\ 3 \cdot p + 4 \cdot r & 3 \cdot q + 4 \cdot s \end{pmatrix} = \begin{pmatrix} p + 2r & q + 2s \\ 3p + 4r & 3q + 4s \end{pmatrix} \]

### Step 3: Set the equations equal
Since \( AB = BA \), we equate the two resulting matrices:
\[ \begin{pmatrix} p + 3q & 2p + 4q \\ r + 3s & 2r + 4s \end{pmatrix} = \begin{pmatrix} p + 2r & q + 2s \\ 3p + 4r & 3q + 4s \end{pmatrix} \]

This gives us the following equations:
1. \( p + 3q = p + 2r \)
2. \( 2p + 4q = q + 2s \)
3. \( r + 3s = 3p + 4r \)
4. \( 2r + 4s = 3q + 4s \)

### Step 4: Simplify the equations
From the first equation:
\[ 3q = 2r \quad \Rightarrow \quad r = \frac{3}{2}q \]

From the second equation:
\[ 2p + 4q = q + 2s \quad \Rightarrow \quad 2p + 3q = 2s \quad \Rightarrow \quad s = p + \frac{3}{2}q \]

From the third equation:
\[ r + 3s = 3p + 4r \quad \Rightarrow \quad 3s = 3p + 3r \quad \Rightarrow \quad s = p + r \]

Substituting \( r = \frac{3}{2}q \) into \( s = p + r \):
\[ s = p + \frac{3}{2}q \]

From the fourth equation:
\[ 2r + 4s = 3q + 4s \quad \Rightarrow \quad 2r = 3q \quad \Rightarrow \quad r = \frac{3}{2}q \]
This is consistent with our previous findings.

### Conclusion
The general form of matrix \( A \) can be expressed as:
\[ A = \begin{pmatrix} p & q \\ \frac{3}{2}q & p + \frac{3}{2}q \end{pmatrix} \]
where \( p \) and \( q \) can be any real numbers.