If a, b and c are any three vectors and their inverse are `a^(-1), b^(-1) and c^(-1) and [a b c]ne0`, then `[a^(-1) b^(-1) c^(-1)]` will be

Options

A. zero
B. one
C. non-zero
D. [a b c]

Answer

To solve the problem, we need to find the scalar triple product of the inverses of three vectors \( a, b, c \), given that the scalar triple product of \( a, b, c \) is non-zero.

### Step-by-Step Solution:

1. **Understanding Scalar Triple Product**:
The scalar triple product of three vectors \( a, b, c \) is given by the determinant of the matrix formed by these vectors. It can be denoted as \( [a \, b \, c] \).

2. **Property of Inverses**:
There is a property relating the scalar triple product of three vectors and their inverses. Specifically, for any three vectors \( a, b, c \):
\[ [a^{-1} \, b^{-1} \, c^{-1}] = \frac{1}{[a \, b \, c]} \]
This means that the scalar triple product of the inverses is the reciprocal of the scalar triple product of the original vectors.

3. **Given Condition**:
We know from the problem statement that \( [a \, b \, c] \neq 0 \). This indicates that the vectors \( a, b, c \) are not coplanar and thus the scalar triple product is non-zero.

4. **Applying the Property**:
Using the property mentioned above, we can substitute into our equation:
\[ [a^{-1} \, b^{-1} \, c^{-1}] = \frac{1}{[a \, b \, c]} \]
Since \( [a \, b \, c] \neq 0 \), it follows that \( \frac{1}{[a \, b \, c]} \) is also non-zero.

5. **Conclusion**:
Therefore, we conclude that:
\[ [a^{-1} \, b^{-1} \, c^{-1}] \neq 0 \]
This means that the scalar triple product of the inverses of the vectors is also non-zero.

### Final Answer:
The scalar triple product \( [a^{-1} \, b^{-1} \, c^{-1}] \) is non-zero.