If the matrix P = `[(7,a,4),(-1,3,b),(c, 6, 2)]` is a symmetric matrix then the respective values of a ,b,c are

Options

A. 4,6,-1
B. 6,4,-1
C. `-1`, 6,4
D. 6,-1,4

Answer

To find the values of \( a \), \( b \), and \( c \) in the symmetric matrix \( P = \begin{pmatrix} 7 & a & 4 \\ -1 & 3 & b \\ c & 6 & 2 \end{pmatrix} \), we will use the property of symmetric matrices, which states that a matrix is symmetric if its transpose is equal to the original matrix.

### Step-by-Step Solution:

1. **Transpose the Matrix \( P \)**:
The transpose of a matrix is obtained by swapping its rows and columns. Therefore, the transpose of matrix \( P \) is:
\[ P^T = \begin{pmatrix} 7 & -1 & c \\ a & 3 & 6 \\ 4 & b & 2 \end{pmatrix} \]

2. **Set the Transpose Equal to the Original Matrix**:
Since \( P \) is symmetric, we have:
\[ P^T = P \]
This gives us:
\[ \begin{pmatrix} 7 & -1 & c \\ a & 3 & 6 \\ 4 & b & 2 \end{pmatrix} = \begin{pmatrix} 7 & a & 4 \\ -1 & 3 & b \\ c & 6 & 2 \end{pmatrix} \]

3. **Equate Corresponding Elements**:
From the equality of the two matrices, we can equate the corresponding elements:
- From the first row: \( a = -1 \) and \( 4 = c \)
- From the second row: \( b = 6 \)

4. **Write the Values**:
From the equations we derived:
- \( a = -1 \)
- \( b = 6 \)
- \( c = 4 \)

Thus, the respective values of \( a \), \( b \), and \( c \) are:
- \( a = -1 \)
- \( b = 6 \)
- \( c = 4 \)

### Final Answer:
- \( a = -1 \)
- \( b = 6 \)
- \( c = 4 \)