If `A=[(5,x),( y,0)]` and `A=A^T` , then

Options

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

Explanation

To solve the problem, we start with the given matrix \( A \) and the condition that \( A = A^T \).

### Step 1: Write down the matrix \( A \)
The matrix \( A \) is given as:
\[ A = \begin{pmatrix} 5 & x \\ y & 0 \end{pmatrix} \]

### Step 2: Write down the transpose of matrix \( A \)
The transpose of matrix \( A \), denoted as \( A^T \), is obtained by swapping the rows and columns:
\[ A^T = \begin{pmatrix} 5 & y \\ x & 0 \end{pmatrix} \]

### Step 3: Set the matrices equal to each other
Since we know that \( A = A^T \), we can set the two matrices equal:
\[ \begin{pmatrix} 5 & x \\ y & 0 \end{pmatrix} = \begin{pmatrix} 5 & y \\ x & 0 \end{pmatrix} \]

### Step 4: Equate corresponding elements
From the equality of the matrices, we can equate the corresponding elements:
1. From the first row, first column: \( 5 = 5 \) (this is always true)
2. From the first row, second column: \( x = y \)
3. From the second row, first column: \( y = x \)
4. From the second row, second column: \( 0 = 0 \) (this is always true)

### Step 5: Conclusion
From the equations derived, we have:
\[ x = y \]
This means that the values of \( x \) and \( y \) are equal.

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