Let `A=[(1,2),(3,-5)],B=[(1,0),(0,2)]` and X be a matrix such that A=BX, then X is equal to:

Options

A. `1/2[(2,4),(3,-5)]`
B. `1/2[(-2,4),(3,5)]`
C. `[(2,4),(3,-5)]`
D. None of these

Answer

To find the matrix \( X \) such that \( A = BX \), we can express \( X \) as \( X = A B^{-1} \). We will follow these steps to find \( X \).

### Step 1: Define the matrices
Given:
\[ A = \begin{pmatrix} 1 & 2 \\ 3 & -5 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} \]

### Step 2: Calculate the determinant of matrix \( B \)
The determinant of matrix \( B \) is calculated as follows:
\[ \text{det}(B) = (1)(2) - (0)(0) = 2 \]

### Step 3: Calculate the adjoint of matrix \( B \)
The adjoint of matrix \( B \) is obtained by swapping the diagonal elements and changing the sign of the off-diagonal elements:
\[ \text{adj}(B) = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix} \]

### Step 4: Calculate the inverse of matrix \( B \)
The inverse of matrix \( B \) is given by:
\[ B^{-1} = \frac{1}{\text{det}(B)} \cdot \text{adj}(B) = \frac{1}{2} \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & \frac{1}{2} \end{pmatrix} \]

### Step 5: Calculate \( X = A B^{-1} \)
Now we multiply matrix \( A \) with \( B^{-1} \):
\[ X = A B^{-1} = \begin{pmatrix} 1 & 2 \\ 3 & -5 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & \frac{1}{2} \end{pmatrix} \]
Calculating the product:
\[ X = \begin{pmatrix} 1 \cdot 1 + 2 \cdot 0 & 1 \cdot 0 + 2 \cdot \frac{1}{2} \\ 3 \cdot 1 + (-5) \cdot 0 & 3 \cdot 0 + (-5) \cdot \frac{1}{2} \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 3 & -\frac{5}{2} \end{pmatrix} \]

### Final Result
Thus, the matrix \( X \) is:
\[ X = \begin{pmatrix} 1 & 1 \\ 3 & -\frac{5}{2} \end{pmatrix} \]