Let `A=[(1, 2),(3, 4)] and B=[(p,q),(r,s)]` are two matrices such that AB = BA and `r ne 0`, then the value of `(3p-3s)/(5q-4r)` is equal to

Options

A. `(3)/(2)`
B. 4
C. `(9)/(2)`
D. 5

Answer

To solve the problem, we need to find the value of the expression \((3p - 3s) / (5q - 4r)\) given that the matrices \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and \(B = \begin{pmatrix} p & q \\ r & s \end{pmatrix}\) commute (i.e., \(AB = BA\)) and \(r \neq 0\).

### Step-by-Step Solution:

1. **Matrix Multiplication**:
We start by calculating \(AB\) and \(BA\):
\[ AB = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} p & q \\ r & s \end{pmatrix} = \begin{pmatrix} 1p + 2r & 1q + 2s \\ 3p + 4r & 3q + 4s \end{pmatrix} \]
\[ BA = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} p + 3q & 2p + 4q \\ r + 3s & 2r + 4s \end{pmatrix} \]

2. **Setting Up the Equations**:
Since \(AB = BA\), we can set the corresponding elements equal to each other:
- From the first element:
\[ p + 2r = p + 3q \implies 2r = 3q \implies q = \frac{2r}{3} \]
- From the second element:
\[ 1q + 2s = 2p + 4q \implies q + 2s = 2p + 4q \implies 2s - 2p = 3q \implies 2s = 2p + 3q \]
- Substitute \(q\) from the first equation:
\[ 2s = 2p + 3\left(\frac{2r}{3}\right) \implies 2s = 2p + 2r \implies s = p + r \]
- From the third element:
\[ 3p + 4r = r + 3s \implies 3p + 4r = r + 3(p + r) \implies 3p + 4r = r + 3p + 3r \implies 4r = 4r \]
- This equation is always true, confirming our previous results.

3. **Finding the Expression**:
Now we need to calculate \((3p - 3s) / (5q - 4r)\):
- Substitute \(s = p + r\) into \(3p - 3s\):
\[ 3p - 3s = 3p - 3(p + r) = 3p - 3p - 3r = -3r \]
- Substitute \(q = \frac{2r}{3}\) into \(5q - 4r\):
\[ 5q - 4r = 5\left(\frac{2r}{3}\right) - 4r = \frac{10r}{3} - 4r = \frac{10r}{3} - \frac{12r}{3} = -\frac{2r}{3} \]

4. **Final Calculation**:
Now substituting back into the expression:
\[ \frac{3p - 3s}{5q - 4r} = \frac{-3r}{-\frac{2r}{3}} = \frac{-3r \cdot 3}{-2r} = \frac{9}{2} \]

### Conclusion:
The value of \(\frac{3p - 3s}{5q - 4r}\) is \(\frac{9}{2}\).