Let a, b and c are the roots of the equation `x^(3)-7x^(2)+9x-13=0` and A and B are two matrices given by `A=[(a,b,c),(b,c,a),(c,a,b)] and B=[(bc-a^(2),ca-b^(2),ab-c^(2)),(ca-b^(2),ab-c^(2),bc-a^(2)),(ab-c^(2),bc-a^(2),ca-b^(2))]`, then the value `|A||B|` is equal to

Options

A. `-154`
B. `-(154)^(3)`
C. `-22`
D. `-(22)^(3)`

Answer

To solve the problem, we need to find the value of the product of the determinants of matrices A and B, denoted as \(|A||B|\). We start by analyzing the given polynomial and the matrices.

### Step 1: Identify the roots of the polynomial
The polynomial given is:
\[ x^3 - 7x^2 + 9x - 13 = 0 \]
Let \(a\), \(b\), and \(c\) be the roots of this polynomial. By Vieta's formulas, we know:
- \(a + b + c = 7\)
- \(ab + bc + ca = 9\)
- \(abc = 13\)

### Step 2: Formulate matrix A
The matrix \(A\) is given by:
\[ A = \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix} \]

### Step 3: Calculate the determinant of matrix A
To find the determinant of matrix \(A\), we can use the formula for the determinant of a 3x3 matrix:
\[ |A| = a \begin{vmatrix} c & a \\ a & b \end{vmatrix} - b \begin{vmatrix} b & a \\ c & b \end{vmatrix} + c \begin{vmatrix} b & c \\ c & a \end{vmatrix} \]

Calculating the minors:
1. \(\begin{vmatrix} c & a \\ a & b \end{vmatrix} = cb - a^2\)
2. \(\begin{vmatrix} b & a \\ c & b \end{vmatrix} = bb - ac = b^2 - ac\)
3. \(\begin{vmatrix} b & c \\ c & a \end{vmatrix} = ba - c^2\)

Substituting these into the determinant formula:
\[ |A| = a(cb - a^2) - b(b^2 - ac) + c(ba - c^2) \]
Expanding this gives:
\[ |A| = acb - a^3 - b^3 + abc + abc - c^3 \]
Combining like terms:
\[ |A| = 3abc - (a^3 + b^3 + c^3) \]

### Step 4: Use the identity for \(a^3 + b^3 + c^3\)
We can use the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)((a + b + c)^2 - 3(ab + ac + bc)) \]
Substituting the known values:
\[ a^3 + b^3 + c^3 = 3abc + (a + b + c)((a + b + c)^2 - 3(ab + ac + bc)) \]
Calculating:
\[ = 3 \cdot 13 + 7(7^2 - 3 \cdot 9) = 39 + 7(49 - 27) = 39 + 7 \cdot 22 = 39 + 154 = 193 \]

### Step 5: Substitute back to find \(|A|\)
Now substituting back into the determinant:
\[ |A| = 3 \cdot 13 - 193 = 39 - 193 = -154 \]

### Step 6: Calculate the determinant of matrix B
Since matrix \(B\) is the cofactor matrix of \(A\), we have:
\[ |B| = |A|^2 \]
Thus:
\[ |B| = (-154)^2 = 23716 \]

### Step 7: Find \(|A||B|\)
Now we can find:
\[ |A||B| = |A| \cdot |B| = (-154) \cdot 23716 = -3652034 \]

### Final Answer
The value of \(|A||B|\) is:
\[ \boxed{-3652034} \]