If A and B are invertible matrices of the same order, then `(AB)^(-1)` is equal to :

Options

A. `BA`
B. `B^(-1)A`
C. `BA^(-1)`
D. `B^(-1)A^(-1)`.

Answer

To find the inverse of the product of two invertible matrices \( A \) and \( B \), we can use the property of matrix inverses. The formula we want to prove is:

\[ (AB)^{-1} = B^{-1} A^{-1} \]

### Step-by-Step Solution:

1. **Start with the product \( AB \)**:
We know that \( A \) and \( B \) are both invertible matrices of the same order.

2. **Multiply both sides by \( (AB)^{-1} \)**:
We can write:
\[ (AB)(AB)^{-1} = I \]
where \( I \) is the identity matrix.

3. **Pre-multiply by \( B^{-1} \)**:
To isolate \( A \), we can pre-multiply both sides by \( B^{-1} \):
\[ B^{-1}(AB)(AB)^{-1} = B^{-1}I \]
This simplifies to:
\[ (B^{-1}A)(B) = B^{-1} \]

4. **Use the property of identity**:
Since \( B^{-1}B = I \), we have:
\[ B^{-1}A = I \]

5. **Now pre-multiply by \( A^{-1} \)**:
Next, we can pre-multiply both sides by \( A^{-1} \):
\[ A^{-1}(B^{-1}A) = A^{-1}I \]
This simplifies to:
\[ (A^{-1}B^{-1})A = A^{-1} \]

6. **Combine results**:
Since \( A^{-1}A = I \), we can conclude:
\[ A^{-1}B^{-1} = (AB)^{-1} \]

7. **Final result**:
Therefore, we have:
\[ (AB)^{-1} = B^{-1}A^{-1} \]

### Conclusion:
The inverse of the product of two invertible matrices \( A \) and \( B \) is given by:
\[ (AB)^{-1} = B^{-1}A^{-1} \]