The determinant is equal to zero, if :

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Introduction to Determinants

Properties of Determinant

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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 :

a, b, c are in GP

a, b, c are in AP

(x – α) is a factor of ax² + 2bx + c

α is a root of the equation ax² + bx + c = 0

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$∣ \begin{array}{ccc} a & b & aα + b \\ b & c & bα + c \\ aα + b & bα + c & 0 \end{array} ∣ = 0$

By expanding matrix,

a(c(0) – (bα + c) (bα + c)) – b(b(0) – (aα + b) (bα + c)) + (aα + b)(b(bα + c) – c(aα + b)) = 0

(aα + b)(b²α + bc – acα – bc) = a(bα + c)² – b(aα + b)(bα + c)

(aα + b)b²α – acα = (a(bα + c) – b(aα + b))(bα + c)

(aα + b)b² – acα = abα + ac – baα + b²(bα + c)

(aα + b)b² – acα = ac – b²(bα + c)

(aα + b)( –1)α = (bα + c)

– aα² – bα = bα + c

aα² + 2bα + c = 0

Hence, (x – α) is a factor of ax² + 2bx + c