Assertion (A) : If the matrix ` P = [(0,2b,-2),(3,1,3),(3a,3,3)]` is a symmetric matrix, then ` a = (-2)/(3) and b = (3)/(2)`
Reason (R) : If P is a symmetric matric, then p' = - p

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false and R is true

Answer

To solve the problem, we need to determine the values of \( a \) and \( b \) for the matrix \( P = \begin{pmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & 3 \end{pmatrix} \) to be symmetric. A matrix is symmetric if it is equal to its transpose, i.e., \( P = P^T \).

### Step-by-step Solution:

1. **Find the Transpose of Matrix \( P \)**:
The transpose of a matrix is obtained by swapping its rows with columns. Therefore, the transpose \( P^T \) of matrix \( P \) is:
\[ P^T = \begin{pmatrix} 0 & 3 & 3a \\ 2b & 1 & 3 \\ -2 & 3 & 3 \end{pmatrix} \]

2. **Set Up the Equation \( P = P^T \)**:
For \( P \) to be symmetric, we need:
\[ \begin{pmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & 3 \end{pmatrix} = \begin{pmatrix} 0 & 3 & 3a \\ 2b & 1 & 3 \\ -2 & 3 & 3 \end{pmatrix} \]

3. **Compare Corresponding Elements**:
We will compare the elements of both matrices:
- From the first row:
- \( 0 = 0 \) (True)
- \( 2b = 3 \)
- \( -2 = 3a \)

- From the second row:
- \( 3 = 2b \)
- \( 1 = 1 \) (True)
- \( 3 = 3 \) (True)

- From the third row:
- \( 3a = -2 \)
- \( 3 = 3 \) (True)
- \( 3 = 3 \) (True)

4. **Solve the Equations**:
- From \( 2b = 3 \):
\[ b = \frac{3}{2} \]

- From \( -2 = 3a \):
\[ 3a = -2 \implies a = \frac{-2}{3} \]

5. **Conclusion**:
The values of \( a \) and \( b \) that make the matrix \( P \) symmetric are:
\[ a = \frac{-2}{3}, \quad b = \frac{3}{2} \]