First slide

Introduction to Determinants

Question

The determinant $|\begin{array}{ccc} a & b & aα + b \\ b & c & bα + c \\ aα + b & bα + c & 0 \end{array}| = 0$ is equal to zero, if

Moderate

A

a, b, c are in A.P.
B

a, b, c are in G.P.
C

$α$ is a root of the equation ${ax}^{2} + bx + c = 0$
D

$(x - α)$ is a factor of ${ax}^{2} + 2 bx + c$

Solution

Given that $|\begin{array}{ccc} a & b & aα + b \\ b & c & bα + c \\ aα + b & bα + c & 0 \end{array}| = 0$
Operating $C_{3} \to C_{3} - C_{1} α - C_{2}$ , we get
$|\begin{array}{ccc} a & b & 0 \\ b & c & 0 \\ aα + b & bα + c & - ({aα}^{2} + bα + bα + c) \end{array}| = 0$
$\begin{array}{l} \Rightarrow ({aα}^{2} + 2 bα + c) |\begin{array}{ccc} a & b & 0 \\ b & c & 0 \\ aα + b & bα + c & 1 \end{array}| = 0 \\ \Rightarrow (ac - b^{2}) ({aα}^{2} + 2 bα + c) = 0 \end{array}$
So, either $ac - b^{2} = 0 or {aα}^{2} + 2 bα + c = 0$
This means that either a, b, c are in G.P. or $(x - α)$ is a factor of ${ax}^{2} + 2 bx + c$ .

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