For a square matrix A, which of the following properties hold ?
1. `(A^(-1))^(-1)=A`
2. `det(A^(-1))=(1)/(det A)`
3. `(lambda A)^(-1)lambda A^(-1)` where `lambda` is a scalar
Select the correct answer using the code given below :

Options

A. 1 and 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1, 2 and 3

Explanation

To determine which properties hold for a square matrix \( A \), let's analyze each property step by step.

### Step 1: Analyze the first property
**Property 1:** \((A^{-1})^{-1} = A\)

- The inverse of a matrix \( A \) is defined such that when you multiply \( A \) by its inverse \( A^{-1} \), you get the identity matrix \( I \):
\[ A \cdot A^{-1} = I \]
- If you take the inverse of \( A^{-1} \), you should return to the original matrix \( A \):
\[ (A^{-1})^{-1} = A \]
- Therefore, this property is **true**.

### Step 2: Analyze the second property
**Property 2:** \(\text{det}(A^{-1}) = \frac{1}{\text{det}(A)}\)

- The determinant of the inverse of a matrix is given by the formula:
\[ \text{det}(A^{-1}) = \frac{1}{\text{det}(A)} \]
- This property holds true for any invertible square matrix \( A \). Therefore, this property is **true**.

### Step 3: Analyze the third property
**Property 3:** \((\lambda A)^{-1} = \lambda A^{-1}\) where \( \lambda \) is a scalar.

- The correct property states that:
\[ (\lambda A)^{-1} = \frac{1}{\lambda} A^{-1} \]
- This means that when you take the inverse of a scalar multiplied by a matrix, you must also take the reciprocal of the scalar. Therefore, this property is **false**.

### Conclusion
Based on the analysis:
- Property 1 is true.
- Property 2 is true.
- Property 3 is false.

Thus, the correct answer is **Option A: 1 and 2 only**.