If `A=[(1,2,2),(2,1,2),(2,2,1)],` then the value of `|A^4-18A^2-32A|` is

Options

A. 1
B. 2
C. 3
D. none of these

Explanation

To solve the problem, we need to calculate the determinant of the expression \( |A^4 - 18A^2 - 32A| \) where \( A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix} \).

### Step 1: Calculate \( A^2 \)

First, we will calculate \( A^2 \) by multiplying matrix \( A \) by itself.

\[ A^2 = A \cdot A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix} \]

Calculating each element of \( A^2 \):

- First row, first column: \( 1 \cdot 1 + 2 \cdot 2 + 2 \cdot 2 = 1 + 4 + 4 = 9 \)
- First row, second column: \( 1 \cdot 2 + 2 \cdot 1 + 2 \cdot 2 = 2 + 2 + 4 = 8 \)
- First row, third column: \( 1 \cdot 2 + 2 \cdot 2 + 2 \cdot 1 = 2 + 4 + 2 = 8 \)

- Second row, first column: \( 2 \cdot 1 + 1 \cdot 2 + 2 \cdot 2 = 2 + 2 + 4 = 8 \)
- Second row, second column: \( 2 \cdot 2 + 1 \cdot 1 + 2 \cdot 2 = 4 + 1 + 4 = 9 \)
- Second row, third column: \( 2 \cdot 2 + 1 \cdot 2 + 2 \cdot 1 = 4 + 2 + 2 = 8 \)

- Third row, first column: \( 2 \cdot 1 + 2 \cdot 2 + 1 \cdot 2 = 2 + 4 + 2 = 8 \)
- Third row, second column: \( 2 \cdot 2 + 2 \cdot 1 + 1 \cdot 2 = 4 + 2 + 2 = 8 \)
- Third row, third column: \( 2 \cdot 2 + 2 \cdot 2 + 1 \cdot 1 = 4 + 4 + 1 = 9 \)

Thus,

\[ A^2 = \begin{pmatrix} 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 \end{pmatrix} \]

### Step 2: Calculate \( A^4 \)

Next, we calculate \( A^4 \) as \( A^2 \cdot A^2 \):

\[ A^4 = A^2 \cdot A^2 = \begin{pmatrix} 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 \end{pmatrix} \cdot \begin{pmatrix} 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 \end{pmatrix} \]

Calculating each element of \( A^4 \):

- First row, first column: \( 9 \cdot 9 + 8 \cdot 8 + 8 \cdot 8 = 81 + 64 + 64 = 209 \)
- First row, second column: \( 9 \cdot 8 + 8 \cdot 9 + 8 \cdot 8 = 72 + 72 + 64 = 208 \)
- First row, third column: \( 9 \cdot 8 + 8 \cdot 8 + 8 \cdot 9 = 72 + 64 + 72 = 208 \)

- Second row, first column: \( 8 \cdot 9 + 9 \cdot 8 + 8 \cdot 8 = 72 + 72 + 64 = 208 \)
- Second row, second column: \( 8 \cdot 8 + 9 \cdot 9 + 8 \cdot 8 = 64 + 81 + 64 = 209 \)
- Second row, third column: \( 8 \cdot 8 + 9 \cdot 8 + 8 \cdot 9 = 64 + 72 + 72 = 208 \)

- Third row, first column: \( 8 \cdot 9 + 8 \cdot 8 + 9 \cdot 8 = 72 + 64 + 72 = 208 \)
- Third row, second column: \( 8 \cdot 8 + 8 \cdot 9 + 9 \cdot 8 = 64 + 72 + 72 = 208 \)
- Third row, third column: \( 8 \cdot 8 + 8 \cdot 8 + 9 \cdot 9 = 64 + 64 + 81 = 209 \)

Thus,

\[ A^4 = \begin{pmatrix} 209 & 208 & 208 \\ 208 & 209 & 208 \\ 208 & 208 & 209 \end{pmatrix} \]

### Step 3: Calculate \( 18A^2 \)

Now we calculate \( 18A^2 \):

\[ 18A^2 = 18 \cdot \begin{pmatrix} 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 \end{pmatrix} = \begin{pmatrix} 162 & 144 & 144 \\ 144 & 162 & 144 \\ 144 & 144 & 162 \end{pmatrix} \]

### Step 4: Calculate \( 32A \)

Next, we calculate \( 32A \):

\[ 32A = 32 \cdot \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix} = \begin{pmatrix} 32 & 64 & 64 \\ 64 & 32 & 64 \\ 64 & 64 & 32 \end{pmatrix} \]

### Step 5: Calculate \( A^4 - 18A^2 - 32A \)

Now we can compute the expression \( A^4 - 18A^2 - 32A \):

\[ A^4 - 18A^2 - 32A = \begin{pmatrix} 209 & 208 & 208 \\ 208 & 209 & 208 \\ 208 & 208 & 209 \end{pmatrix} - \begin{pmatrix} 162 & 144 & 144 \\ 144 & 162 & 144 \\ 144 & 144 & 162 \end{pmatrix} - \begin{pmatrix} 32 & 64 & 64 \\ 64 & 32 & 64 \\ 64 & 64 & 32 \end{pmatrix} \]

Calculating each element:

- First row, first column: \( 209 - 162 - 32 = 15 \)
- First row, second column: \( 208 - 144 - 64 = 0 \)
- First row, third column: \( 208 - 144 - 64 = 0 \)

- Second row, first column: \( 208 - 144 - 64 = 0 \)
- Second row, second column: \( 209 - 162 - 32 = 15 \)
- Second row, third column: \( 208 - 144 - 64 = 0 \)

- Third row, first column: \( 208 - 144 - 64 = 0 \)
- Third row, second column: \( 208 - 144 - 64 = 0 \)
- Third row, third column: \( 209 - 162 - 32 = 15 \)

Thus,

\[ A^4 - 18A^2 - 32A = \begin{pmatrix} 15 & 0 & 0 \\ 0 & 15 & 0 \\ 0 & 0 & 15 \end{pmatrix} \]

### Step 6: Calculate the Determinant

Now we can calculate the determinant of the resulting matrix:

\[ |A^4 - 18A^2 - 32A| = |15I| = 15^3 = 3375 \]

### Final Answer

The value of \( |A^4 - 18A^2 - 32A| \) is \( 3375 \).