If `A^(2) - 3 A + 2I = 0,` then A is equal to

Options

A. `I`
B. `2I`
C. `[[3,-2],[1,0]]`
D. `[[3,1],[-2,0]]`

Explanation

To solve the equation \( A^2 - 3A + 2I = 0 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ A^2 - 3A + 2I = 0 \]
Rearranging gives:
\[ A^2 - 3A = -2I \]

### Step 2: Factoring the Left Side
Next, we can factor the left-hand side:
\[ A^2 - 3A + 2I = (A - 2I)(A - I) = 0 \]

### Step 3: Setting Each Factor to Zero
For the product to be zero, at least one of the factors must be zero. Thus, we set each factor to zero:
1. \( A - 2I = 0 \)
2. \( A - I = 0 \)

### Step 4: Solving Each Equation
From the first equation:
\[ A - 2I = 0 \implies A = 2I \]
From the second equation:
\[ A - I = 0 \implies A = I \]

### Step 5: Conclusion
The solutions for \( A \) are:
\[ A = 2I \quad \text{or} \quad A = I \]

### Final Answer
Thus, \( A \) can be equal to either \( 2I \) or \( I \).

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