Show that

Engineering

Mathematics

Properties of Determinant

Multiplication of Determinants

maximum and minimum of quadratic and rational expression

Question

Show that $|\begin{array}{ccc} - bc & ca + ab & ca + ab \\ ab + bc & - ca & ab + bc \\ bc + ca & bc + ca & - ab \end{array}| = {(\sum ab)}^{3}$

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given

= ∣ \begin{array}{ccc} - bc & ab + ac & ac + ab \\ ab + bc & - ac & bc + ab \\ ca + bc & bc + ac & - ab \end{array} ∣

= ∣ \begin{array}{ccc} ab + bc + ca & ab + bc + ca & ab + bc + ca \\ ab + bc & - ca & bc + ab \\ ca + bc & bc + ac & - ab \end{array} ∣

[Here

R_{1}^{1} = R_{1} + R_{2} + R_{3}]

[taking (ab + bc + ca) common from first Row]

= (ab + bc + ca) ∣ \begin{array}{ccc} 0 & 0 & 1 \\ ab + bc + ca & - (ab + bc + ca) & bc + ab \\ 0 & ab + bc + ca & - ab \end{array} ∣

[Here C₁ = C₁ – C₂, C₂ = C₂ – C₃]

= (ab + bc + ca) ∣ \begin{array}{cc} ab + bc + ca & - (ab + bc + ca) \\ 0 & ab + bc + ca \end{array} ∣

[ expanding by 1st Row]

= (ab + bc + ca) (ab + bc + ca)²

= (ab + bc + ca)³

= (\sum ab)^{3}

Proved