If A,B,C are non - singular matrices of same order then `(AB^(-1)C)^(-1)=`

Options

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

Answer

To solve the problem, we need to find the inverse of the matrix expression \( (AB^{-1}C)^{-1} \) where \( A, B, C \) are non-singular matrices of the same order.

### Step-by-step Solution:

1. **Understanding the Expression**:
We start with the expression \( (AB^{-1}C)^{-1} \). We need to apply the property of inverses for products of matrices.

**Hint**: Recall that the inverse of a product of matrices can be expressed as the product of their inverses in reverse order.

2. **Applying the Inverse Property**:
Using the property \( (XYZ)^{-1} = Z^{-1}Y^{-1}X^{-1} \), we can rewrite our expression:
\[ (AB^{-1}C)^{-1} = C^{-1}(B^{-1})^{-1}A^{-1} \]

**Hint**: Remember that \( (B^{-1})^{-1} = B \).

3. **Simplifying the Expression**:
Now substituting \( (B^{-1})^{-1} \) with \( B \):
\[ (AB^{-1}C)^{-1} = C^{-1}BA^{-1} \]

**Hint**: Make sure to keep track of the order of multiplication when substituting.

4. **Final Result**:
Thus, we have:
\[ (AB^{-1}C)^{-1} = C^{-1}BA^{-1} \]

This matches with one of the options provided.

### Conclusion:
The final answer is:
\[ (AB^{-1}C)^{-1} = C^{-1}BA^{-1} \]

### Summary of Steps:
1. Recognize the expression to be inverted.
2. Apply the property of inverses for products.
3. Simplify the expression using the property of inverses.
4. Write down the final result.