If `A=[{:(3,-x),(y,5):}]` and A=A' then

Options

A. `x=5,y=-3`
B. x=-3,y=5
C. x=y
D. none of these

Answer

To solve the problem, we need to analyze the given matrix \( A \) and its transpose \( A' \). The matrix \( A \) is defined as:

\[ A = \begin{pmatrix} 3 & -x \\ y & 5 \end{pmatrix} \]

Since it is given that \( A = A' \), we can express the transpose of \( A \):

\[ A' = \begin{pmatrix} 3 & y \\ -x & 5 \end{pmatrix} \]

### Step 1: Set the matrices equal to each other

Since \( A = A' \), we can write:

\[ \begin{pmatrix} 3 & -x \\ y & 5 \end{pmatrix} = \begin{pmatrix} 3 & y \\ -x & 5 \end{pmatrix} \]

### Step 2: Compare corresponding elements

From the equality of the matrices, we can compare the corresponding elements:

1. From the first row, first column:
\[ 3 = 3 \quad \text{(This is always true)} \]

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:
\[ 5 = 5 \quad \text{(This is also always true)} \]

### Step 3: Solve the equations

From the equations \( -x = y \) and \( y = -x \), we can see that both equations are consistent with each other. This implies that:

\[ y = -x \]

### Step 4: Analyze the implications

Since \( y = -x \), we can conclude that:

- If \( x = 0 \), then \( y = 0 \).
- If \( x \neq 0 \), then \( y \) will be the negative of \( x \).

### Conclusion

Thus, the relationship between \( x \) and \( y \) is that they are negatives of each other. Therefore, we can summarize that:

\[ y = -x \]