Question

If $[a\times bb\times cc\times a]=\lambda {\left[abc\right]}^2$, then $\lambda$ is equal to

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (B)

$1$

$[a\times bb\times cc\times a]$

$(a\times b).\left(\right(b\times c)\times (c\times a\left)\right)$

which is quarable product we know that

$(a\times b)\times (c\times d)=\left[abd\right]c-\left[abc\right]d$

therefore

$(a\times b)\left(\right[bca]c-[bcc]a$

$\left[bcc\right]=0$ because when two roots are equal of as three

Therefore $(a\times b)\left(\right[bca\left]c\right)$

$=(a\times b).c\left[bca\right]$

$=\left[abc\right]\left[bca\right]$

$\because \left[bca\right]=\left[abc\right]=\left[cab\right]$

so, $\left(1\right)[abc{]}^2$

$\lambda =1$.