Let A be a non - singular symmetric matrix of order 3. If `A^(T)=A^(2)-I`, then `(A-I)^(-1)` is equal to

Options

A. A
B. 2A
C. `A-I`
D. `2A-I`

Answer

To solve the problem, we need to find \((A - I)^{-1}\) given that \(A^T = A^2 - I\) for a non-singular symmetric matrix \(A\).

### Step 1: Use the given equation
We start with the equation given in the problem:
\[ A^T = A^2 - I \]
Since \(A\) is symmetric, we have \(A^T = A\). Therefore, we can rewrite the equation as:
\[ A = A^2 - I \]

### Step 2: Rearranging the equation
Rearranging the equation gives us:
\[ A^2 - A - I = 0 \]
This is a quadratic equation in terms of the matrix \(A\).

### Step 3: Factor the quadratic equation
We can factor the quadratic equation:
\[ A^2 - A - I = 0 \implies A^2 - A = I \]
This can be rearranged to:
\[ A(A - I) = I \]

### Step 4: Identify the inverse
From the equation \(A(A - I) = I\), we see that \(A\) is the inverse of \((A - I)\). Therefore, we can express this as:
\[ (A - I)^{-1} = A \]

### Conclusion
Thus, we have found that:
\[ (A - I)^{-1} = A \]