First slide

Introduction to Determinants

Question

The value of the determinant $|\begin{array}{lll} ka k^{2} + a^{2} 1 \\ kb k^{2} + b^{2} 1 \\ kc k^{2} + c^{2} 1 \end{array}|$ is

Moderate

A

$k (a + b) (b + c) (c + a)$
B

$kabc (a^{2} + b^{2} + c^{2})$
C

$k (a - b) (b - c) (c - a)$
D

$k (a + b - c) (b + c - a) (c + a - b)$

Solution

We have,
$\begin{matrix} |\begin{array}{lll} ka & k^{2} + a^{2} & 1 \\ kb & k^{2} + b^{2} & 1 \\ kc & k^{2} + c^{2} & 1 \end{array}| = |\begin{array}{lll} ka & k^{2} & 1 \\ kb & k^{2} & 1 \\ kc & k^{2} & 1 \end{array}| + |\begin{array}{lll} ka & a^{2} & 1 \\ kb & b^{2} & 1 \\ kc & c^{2} & 1 \end{array}| \\ = 0 + k |\begin{array}{lll} a & a^{2} & 1 \\ b & b^{2} & 1 \\ c & c^{2} & 1 \end{array}| \\ = k (a - b) (b - c) (c - a) \end{matrix}$

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