If `[{:(5,0),(0,7):}]^(-1)[{:(x),(-y):}]=[{:(-1),(2):}]`, then which one of the following is correct ?

Options

A. x = 5, y = 14
B. x = -5, y = 15
C. x = -5, y = -14
D. x = 5, y = -14

Explanation

To solve the given problem, we need to find the values of \( x \) and \( y \) from the equation involving matrices. Let's break it down step by step.

### Step 1: Write down the given equation

We are given:

\[ \begin{pmatrix} 5 & 0 \\ 0 & 7 \end{pmatrix}^{-1} \begin{pmatrix} x \\ -y \end{pmatrix} = \begin{pmatrix} -1 \\ 2 \end{pmatrix} \]

### Step 2: Find the inverse of the matrix

The inverse of a diagonal matrix can be found by taking the reciprocal of each of the diagonal elements. Thus, the inverse of the matrix is:

\[ \begin{pmatrix} 5 & 0 \\ 0 & 7 \end{pmatrix}^{-1} = \begin{pmatrix} \frac{1}{5} & 0 \\ 0 & \frac{1}{7} \end{pmatrix} \]

### Step 3: Multiply both sides by the inverse matrix

Now we multiply both sides of the equation by the inverse matrix:

\[ \begin{pmatrix} \frac{1}{5} & 0 \\ 0 & \frac{1}{7} \end{pmatrix} \begin{pmatrix} x \\ -y \end{pmatrix} = \begin{pmatrix} -1 \\ 2 \end{pmatrix} \]

### Step 4: Perform the matrix multiplication

This results in:

\[ \begin{pmatrix} \frac{1}{5}x \\ -\frac{1}{7}y \end{pmatrix} = \begin{pmatrix} -1 \\ 2 \end{pmatrix} \]

### Step 5: Set up equations from the matrix multiplication

From the resulting matrix, we can set up the following equations:

1. \( \frac{1}{5}x = -1 \)
2. \( -\frac{1}{7}y = 2 \)

### Step 6: Solve for \( x \)

From the first equation:

\[ x = -1 \times 5 = -5 \]

### Step 7: Solve for \( y \)

From the second equation:

\[ y = -2 \times 7 = -14 \]

### Conclusion

Thus, we find:

\[ x = -5, \quad y = -14 \]

### Final Answer

The correct option is \( x = -5 \) and \( y = -14 \).

---