If A and B are square matrices of order 3 such that `A^(3)=8 B^(3)=8I` and det. `(AB-A-2B+2I) ne 0`, then identify the correct statement(s), where `I` is identity matrix of order 3.

Options

A. `A^(2)+2A+4I=O`
B. `A^(2)+2A+4I neO`
C. `B^(2)+B+I=O`
D. `B^(2)+B+I ne O`

Explanation

To solve the given problem step by step, we will analyze the conditions provided and derive the necessary equations.

### Step 1: Analyze the given conditions
We are given two matrices \( A \) and \( B \) of order 3, with the following conditions:
1. \( A^3 = 8I \)
2. \( B^3 = 8I \)
3. \( \text{det}(AB - A - 2B + 2I) \neq 0 \)

### Step 2: Rewrite the equations for \( A \) and \( B \)
From \( A^3 = 8I \), we can rewrite this as:
\[ A^3 - 8I = 0 \]
This can be factored as:
\[ (A - 2I)(A^2 + 2A + 4I) = 0 \]

Similarly, from \( B^3 = 8I \), we rewrite this as:
\[ B^3 - 8I = 0 \]
This can be factored as:
\[ (B - 2I)(B^2 + 2B + 4I) = 0 \]

### Step 3: Analyze the determinants
Given that \( \text{det}(AB - A - 2B + 2I) \neq 0 \), we can factor the expression:
\[ AB - A - 2B + 2I = (A - 2I)(B - I) \]
Thus, we can say:
\[ \text{det}((A - 2I)(B - I)) \neq 0 \]
This implies that both \( A - 2I \) and \( B - I \) cannot be the zero matrix, otherwise the determinant would be zero.

### Step 4: Conclude the implications
Since \( A - 2I \neq 0 \), it follows that:
\[ A^2 + 2A + 4I = 0 \]
And since \( B - I \neq 0 \), it follows that:
\[ B^2 + B + I = 0 \]

### Step 5: Final statements
Thus, we conclude that:
1. \( A^2 + 2A + 4I = 0 \)
2. \( B^2 + B + I = 0 \)