Determinants
General
Grade 12
Question:
Show that $\begin{vmatrix} a^2 + \lambda^2 & ab + c\lambda & ca - b\lambda \\ ab - c\lambda & b^2 + \lambda^2 & bc + a\lambda \\ ac + b\lambda & bc - a\lambda & c^2 + \lambda^2 \end{vmatrix} \times \begin{vmatrix} \lambda & c & -b \\ -c & \lambda & a \\ b & -a & \lambda \end{vmatrix} = \lambda^3(\lambda^2 + a^2 + b^2 + c^2)^3$
Step-by-Step Solution
Key Concept: General
We observe that the elements in the first determinant are the cofactors of corresponding elements of second determinant. <br> So $\begin{vmatrix} \lambda & c & -b \\ -c & \lambda & a \\ b & -a & \lambda \end{vmatrix}^3 = [\lambda(\lambda^2 + a^2 + b^2 + c^2)]^3 = \lambda^3(\lambda^2 + a^2 + b^2 + c^2)^3$
Correct Answer: A