Let `A` be a `3xx3` matrix such that `A^2-5A+7I=0` then which of the statements is true

Options

A. statement -1 is false, but statement -2 is true,
B. Both statement are false.
C. Both statement are ture.
D. Statement -1 is true, but statement -2 is false.

Explanation

To solve the problem, we are given a \(3 \times 3\) matrix \(A\) that satisfies the equation:

\[ A^2 - 5A + 7I = 0 \]

We need to determine which of the provided statements is true.

### Step 1: Rearranging the Given Equation
We start by rearranging the given equation to express \(A^2\):

\[ A^2 = 5A - 7I \]

### Step 2: Finding \(A^{-1}\)
Next, we multiply both sides of the original equation by \(A^{-1}\):

\[ A^{-1}(A^2 - 5A + 7I) = A^{-1}0 \]

This simplifies to:

\[ A^{-1}A^2 - 5A^{-1}A + 7A^{-1}I = 0 \]

Using the properties of matrices, we know \(A^{-1}A = I\):

\[ A^{-1}A^2 - 5I + 7A^{-1} = 0 \]

Thus, we can rewrite this as:

\[ A - 5I + 7A^{-1} = 0 \]

Rearranging gives:

\[ 7A^{-1} = 5I - A \]

Dividing by 7, we find:

\[ A^{-1} = \frac{1}{7}(5I - A) \]

### Step 3: Verifying Statement 1
From our calculation, we see that:

**Statement 1:** \(A^{-1} = \frac{1}{7}(5I - A)\) is true.

### Step 4: Finding \(A^3\)
Now we will find \(A^3\) using the expression for \(A^2\):

\[ A^3 = A \cdot A^2 = A(5A - 7I) = 5A^2 - 7A \]

Substituting \(A^2\) from earlier:

\[ A^3 = 5(5A - 7I) - 7A = 25A - 35I - 7A = 18A - 35I \]

### Step 5: Verifying Statement 2
Now we check the polynomial \(A^3 - 2A^2 - 3A + I\):

Substituting \(A^3\) and \(A^2\):

\[ A^3 - 2A^2 - 3A + I = (18A - 35I) - 2(5A - 7I) - 3A + I \]

Calculating this step-by-step:

1. Substitute \(A^2\):
\[ = 18A - 35I - 10A + 14I - 3A + I \]
2. Combine like terms:
\[ = (18A - 10A - 3A) + (-35I + 14I + I) \]
\[ = 5A - 20I \]

This shows that:

\[ A^3 - 2A^2 - 3A + I = 5A - 20I \]

Thus, we can factor this as:

\[ = 5(A - 4I) \]

This confirms that **Statement 2** is also true.

### Conclusion
Both statements are true.

### Final Answer
The correct option is that both statements are true.

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