Let A and B are two non - singular matrices of order 3 such that `A+B=2I` and `A^(-1)+B^(-1)=3I`, then AB is equal to (where, I is the identity matrix of order 3)

Options

A. A
B. B
C. `(2I)/(3)`
D. 2I

Answer

To solve the problem, we need to find the product \( AB \) given the equations \( A + B = 2I \) and \( A^{-1} + B^{-1} = 3I \).

### Step-by-step Solution:

1. **Use the given equations**:
We have:
\[ A + B = 2I \tag{1} \]
\[ A^{-1} + B^{-1} = 3I \tag{2} \]

2. **Rewrite equation (2)**:
We know that:
\[ A^{-1} + B^{-1} = B^{-1}A^{-1}(A + B) = 3I \]
Using the property of inverses, we can rewrite equation (2) as:
\[ A^{-1} + B^{-1} = (AB)^{-1}(A + B) \]
Substituting equation (1) into this gives:
\[ (AB)^{-1}(2I) = 3I \]

3. **Multiply both sides by \((AB)\)**:
\[ 2I = 3AB \]

4. **Solve for \( AB \)**:
Dividing both sides by 3:
\[ AB = \frac{2}{3}I \]

Thus, the product \( AB \) is:
\[ AB = \frac{2}{3}I \]

### Final Answer:
\[ AB = \frac{2}{3}I \]