The determinant is divisible by

Engineering

Mathematics

Introduction to Matrix

Introduction to Determinants

Properties of Determinant

Question

The determinant $Δ = ∣ \begin{array}{ccc} a^{2} + x^{2} & ab & ac \\ ab & b^{2} + x^{2} & bc \\ ac & bc & c^{2} + x^{2} \end{array} ∣$ is divisible by

x²

x⁴

x³

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Δ = ∣ \begin{array}{ccc} a^{2} + x^{2} & ab & ac \\ ab & b^{2} + x^{2} & bc \\ ac & bc & c^{2} + x^{2} \end{array} ∣

Expanding it,

= (a² + x²)(x²)(b² + c² + x²) – ab(abx²) + ac(– acx²)
= x²(a²b² + a²c² + a²x² + x²b² + x²c² + x⁴ – a²b² – a²c²)

= x²(a²x² + x²b² + x²c² + x⁴)

= x⁴(a² + b² + c² + x²)

So it is divisible by x, x², x³ and x⁴