If A = `[{:(5,x),(y , 0):}]` and A = A', then

Options

A. x = 0 , y = 5
B. x + y = 5
C. x = y
D. None of these

Answer

To solve the problem, we need to analyze the given matrix \( A \) and the condition that \( A = A' \), where \( A' \) is the transpose of matrix \( A \).

### Step-by-Step Solution:

1. **Define the matrix \( A \)**:
\[ A = \begin{pmatrix} 5 & x \\ y & 0 \end{pmatrix} \]

2. **Find the transpose of matrix \( A \)**:
The transpose of a matrix is obtained by swapping its rows and columns. Therefore, the transpose \( A' \) of matrix \( A \) is:
\[ A' = \begin{pmatrix} 5 & y \\ x & 0 \end{pmatrix} \]

3. **Set the matrices equal to each other**:
According to the problem, we have:
\[ A = A' \]
This gives us:
\[ \begin{pmatrix} 5 & x \\ y & 0 \end{pmatrix} = \begin{pmatrix} 5 & y \\ x & 0 \end{pmatrix} \]

4. **Equate corresponding elements**:
From the equality of the two matrices, we can equate the corresponding elements:
- From the first row, first column:
\[ 5 = 5 \quad \text{(This is always true)} \]
- From the first row, second column:
\[ x = y \]
- From the second row, first column:
\[ y = x \]
- From the second row, second column:
\[ 0 = 0 \quad \text{(This is also always true)} \]

5. **Conclusion**:
From the equations \( x = y \) and \( y = x \), we conclude that:
\[ x = y \]

### Final Answer:
Thus, the relationship between \( x \) and \( y \) is:
\[ x = y \]