Question

If $a,b$ and $c$ are any three vectors and their inverse are $a^{-1},b^{-1},c^{-1}$ and $[a,b,c]\ne 0$ then $[a^{-1},b^{-1},c^{-1}]$ will be

• A. Zero
• B. One
• C. Non-zero
• D. $\left[abc\right]$

Answer 1

The correct option is C Non-zero

We know $a^{-1}=\frac{b\times c}{\left[abc\right]};b^{-1}=\frac{a\times b}{\left[abc\right]};c^{-1}=\frac{c\times a}{\left[abc\right]}$
$\therefore \left[a^{-1}{b}^{-1}{c}^{-1}\right]=\frac{b\times c}{\left[abc\right]}\{\frac{c\times a}{\left[abc\right]}\times \frac{a\times b}{\left[abc\right]}\}$
$=\frac{b\times c}{\left[abc\right]}.\left\{\frac{a}{\left[abc\right]}\right\}=\frac{1}{\left[abc\right]}\ne 0$