First slide

Applications of determinant

Question

If $|\begin{array}{ccc} a^{2} + λ^{2} & ab + cλ & ca - bλ \\ ab - cλ & b^{2} + λ^{2} & bc + aλ \\ ca + bλ & bc - aλ & c^{2} + λ^{2} \end{array}| |\begin{array}{ccc} λ & c & - b \\ - c & λ & a \\ b & - a & λ \end{array}| = {(1 + a^{2} + b^{2} + c^{2})}^{3}$ , then the value of $λ$ is

Moderate

A

8
B

27
C

1
D

-1

Solution

We observe that the elements in the pre-factor are the cofactors of the corresponding elements of the post-factor. Hence,
${|\begin{array}{ccc} λ & c & - b \\ - c & λ & a \\ b & - a & λ \end{array}|}^{3} = {[λ (λ^{2} + a^{2} + b^{2} + c^{2})]}^{3} = {(1 + a^{2} + b^{2} + c^{2})}^{3}$
$\Rightarrow λ = 1$
Alternate solution:
Writing a = 0, b = 0, c = 0on both sides, we get
$λ^{6} λ^{3} = 1 or λ = 1$

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