If `A=[{:(1,-1,1),(0,2,-3),(2,1,0):}]` and `B=(adjA)` and `C=5A`, then find the value of `(|adjB|)/(|C |)`

Options

A. `25`
B. `2`
C. `1`
D. `5`

Explanation

To solve the problem, we need to find the value of \(\frac{|\text{adj} B|}{|C|}\), where \(A = \begin{pmatrix} 1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 0 \end{pmatrix}\), \(B = \text{adj} A\), and \(C = 5A\).

### Step 1: Calculate the Determinant of Matrix A
First, we need to find the determinant of matrix \(A\).

\[ A = \begin{pmatrix} 1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 0 \end{pmatrix} \]

Using the determinant formula for a \(3 \times 3\) matrix:

\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]

where \(A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\).

Substituting the values:

\[ |A| = 1(2 \cdot 0 - (-3) \cdot 1) - (-1)(0 \cdot 0 - (-3) \cdot 2) + 1(0 \cdot 1 - 2 \cdot 2) \]

Calculating each term:

1. \(1(0 + 3) = 3\)
2. \(-(-1)(0 + 6) = 6\)
3. \(1(0 - 4) = -4\)

So,

\[ |A| = 3 + 6 - 4 = 5 \]

### Step 2: Calculate the Determinant of Matrix C
Now, we find the determinant of \(C\):

\[ C = 5A \]

Using the property of determinants, \(|kA| = k^n |A|\), where \(n\) is the order of the matrix (in this case, \(n = 3\)):

\[ |C| = 5^3 |A| = 125 \cdot 5 = 625 \]

### Step 3: Calculate the Determinant of Matrix B
Next, we need to find \(|\text{adj} A|\). The property of the adjoint states that:

\[ |\text{adj} A| = |A|^{n-1} \]

For our \(3 \times 3\) matrix, \(n = 3\):

\[ |\text{adj} A| = |A|^{3-1} = |A|^2 = 5^2 = 25 \]

### Step 4: Calculate the Value of \(\frac{|\text{adj} B|}{|C|}\)
Now we can find \(\frac{|\text{adj} B|}{|C|}\):

\[ \frac{|\text{adj} B|}{|C|} = \frac{|\text{adj} A|}{|C|} = \frac{25}{625} = \frac{1}{25} = 1 \]

### Final Answer
Thus, the value of \(\frac{|\text{adj} B|}{|C|}\) is \(1\).