First slide

Introduction to Determinants

Question

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

Moderate

A

x
B

x²
C

x³
D

none of these

Solution

$Δ = \frac{1}{a} |\begin{array}{ccc} a^{3} + ax & ab & ac \\ a^{2} b & b^{2} + x & bc \\ a^{2} c & bc & c^{2} + x \end{array}|$
Applying $C_{1} \to C_{1} + {bC}_{2} + {cC}_{3}$ and taking $a^{2} + b^{2} + c^{2} + x$ common, we get
$Δ = \frac{1}{a} (a^{2} + b^{2} + c^{2} + x) |\begin{array}{ccc} a & ab & ac \\ b & b^{2} + x & bc \\ c & bc & c^{2} + x \end{array}|$
Applying $C_{2} \to C_{2} - {bC}_{1}$ and $C_{3} \to C_{3} - {cC}_{1}$ , we get
$\begin{matrix} Δ = \frac{1}{a} (a^{2} + b^{2} + c^{2} + x) |\begin{array}{lll} a & 0 & 0 \\ b & x & 0 \\ c & 0 & x \end{array}| \\ = \frac{1}{a} (a^{2} + b^{2} + c^{2} + x) ({ax}^{2}) \\ = x^{2} (a^{2} + b^{2} + c^{2} + x) \end{matrix}$
Thus $∆$ is divisible by x and x².

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