`A=[(3,a,-1),(2,5,c),(b,8,2)]` is symmetric and `B=[(d,3,a),(b-a,e,-2b-c),(-2,6,-f)]` is skew-symmetric, then find AB.

Explanation

To solve the problem, we need to find the product of matrices \( A \) and \( B \) given that \( A \) is symmetric and \( B \) is skew-symmetric.

### Step 1: Determine the values of \( a, b, c, d, e, f \)

**Matrix A:**
Given:
\[ A = \begin{pmatrix} 3 & a & -1 \\ 2 & 5 & c \\ b & 8 & 2 \end{pmatrix} \]
Since \( A \) is symmetric, we have \( A^T = A \). Therefore:
\[ \begin{pmatrix} 3 & 2 & b \\ a & 5 & 8 \\ -1 & c & 2 \end{pmatrix} = \begin{pmatrix} 3 & a & -1 \\ 2 & 5 & c \\ b & 8 & 2 \end{pmatrix} \]

From this, we can equate the corresponding elements:
1. \( a = 2 \)
2. \( b = -1 \)
3. \( c = 8 \)

**Matrix B:**
Given:
\[ B = \begin{pmatrix} d & 3 & a \\ b-a & e & -2b-c \\ -2 & 6 & -f \end{pmatrix} \]
Since \( B \) is skew-symmetric, we have \( B^T = -B \). Therefore:
\[ \begin{pmatrix} d & b-a & -2 \\ 3 & e & 6 \\ a & -2b-c & -f \end{pmatrix} = \begin{pmatrix} -d & -3 & -a \\ -(b-a) & -e & -6 \\ 2 & -(-2b-c) & -(-f) \end{pmatrix} \]

From this, we can equate the corresponding elements:
1. \( d = 0 \)
2. \( e = 0 \)
3. \( f = 0 \)

### Step 2: Substitute the values into matrices A and B

Now substituting the values we found into matrices \( A \) and \( B \):

**Matrix A:**
\[ A = \begin{pmatrix} 3 & 2 & -1 \\ 2 & 5 & 8 \\ -1 & 8 & 2 \end{pmatrix} \]

**Matrix B:**
\[ B = \begin{pmatrix} 0 & 3 & 2 \\ -3 & 0 & -6 \\ -2 & 6 & 0 \end{pmatrix} \]

### Step 3: Calculate the product \( AB \)

To find \( AB \), we multiply the two matrices:
\[ AB = A \cdot B = \begin{pmatrix} 3 & 2 & -1 \\ 2 & 5 & 8 \\ -1 & 8 & 2 \end{pmatrix} \cdot \begin{pmatrix} 0 & 3 & 2 \\ -3 & 0 & -6 \\ -2 & 6 & 0 \end{pmatrix} \]

Calculating each element of the resulting matrix:
1. First row:
- \( (3 \cdot 0) + (2 \cdot -3) + (-1 \cdot -2) = 0 - 6 + 2 = -4 \)
- \( (3 \cdot 3) + (2 \cdot 0) + (-1 \cdot 6) = 9 + 0 - 6 = 3 \)
- \( (3 \cdot 2) + (2 \cdot -6) + (-1 \cdot 0) = 6 - 12 + 0 = -6 \)

2. Second row:
- \( (2 \cdot 0) + (5 \cdot -3) + (8 \cdot -2) = 0 - 15 - 16 = -31 \)
- \( (2 \cdot 3) + (5 \cdot 0) + (8 \cdot 6) = 6 + 0 + 48 = 54 \)
- \( (2 \cdot 2) + (5 \cdot -6) + (8 \cdot 0) = 4 - 30 + 0 = -26 \)

3. Third row:
- \( (-1 \cdot 0) + (8 \cdot -3) + (2 \cdot -2) = 0 - 24 - 4 = -28 \)
- \( (-1 \cdot 3) + (8 \cdot 0) + (2 \cdot 6) = -3 + 0 + 12 = 9 \)
- \( (-1 \cdot 2) + (8 \cdot -6) + (2 \cdot 0) = -2 - 48 + 0 = -50 \)

Thus, the product \( AB \) is:
\[ AB = \begin{pmatrix} -4 & 3 & -6 \\ -31 & 54 & -26 \\ -28 & 9 & -50 \end{pmatrix} \]