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 \((3p - 3s)/(5q - 4r)\) given that the matrices \(A\) and \(B\) commute, i.e., \(AB = BA\).

Let's start by calculating the products \(AB\) and \(BA\).

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

Given:
\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \]

The product \(AB\) is calculated as follows:
\[ AB = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} p & q \\ r & s \end{pmatrix} = \begin{pmatrix} 1 \cdot p + 2 \cdot r & 1 \cdot q + 2 \cdot s \\ 3 \cdot p + 4 \cdot r & 3 \cdot q + 4 \cdot s \end{pmatrix} \]
This simplifies to:
\[ AB = \begin{pmatrix} p + 2r & q + 2s \\ 3p + 4r & 3q + 4s \end{pmatrix} \]

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

Now, calculate \(BA\):
\[ BA = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} p \cdot 1 + q \cdot 3 & p \cdot 2 + q \cdot 4 \\ r \cdot 1 + s \cdot 3 & r \cdot 2 + s \cdot 4 \end{pmatrix} \]
This simplifies to:
\[ BA = \begin{pmatrix} p + 3q & 2p + 4q \\ r + 3s & 2r + 4s \end{pmatrix} \]

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

Since \(AB = BA\), we can equate the corresponding elements:

1. From the first element:
\[ p + 2r = p + 3q \implies 2r = 3q \implies r = \frac{3}{2}q \]

2. From the second element:
\[ q + 2s = 2p + 4q \implies 2s - 2p = 3q \implies s - p = \frac{3}{2}q \implies p - s = -\frac{3}{2}q \]

### Step 4: Express \(3p - 3s\)

Now, we can express \(3p - 3s\):
\[ 3p - 3s = 3(p - s) = 3\left(-\frac{3}{2}q\right) = -\frac{9}{2}q \]

### Step 5: Express \(5q - 4r\)

Next, we express \(5q - 4r\):
Using \(r = \frac{3}{2}q\):
\[ 5q - 4r = 5q - 4\left(\frac{3}{2}q\right) = 5q - 6q = -q \]

### Step 6: Compute \(\frac{3p - 3s}{5q - 4r}\)

Now we can substitute these values into the expression:
\[ \frac{3p - 3s}{5q - 4r} = \frac{-\frac{9}{2}q}{-q} = \frac{9}{2} \]

### Final Answer

Thus, the value of \(\frac{3p - 3s}{5q - 4r}\) is:
\[ \boxed{\frac{9}{2}} \]