If and a2 + b2 + c2 = 1 then A2 =

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

If $A = [\begin{array}{lll} a^{2} & ab & ac \\ ab & b^{2} & bc \\ ac & bc & c^{2} \end{array}]$ and a² + b² + c² = 1 then A² =

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Given

A = [\begin{array}{ccc} a^{2} & ab & ac \\ ab & b^{2} & bc \\ ac & bc & c^{2} \end{array}]

A^{2} = A \times A = [\begin{array}{ccc} a^{2} & ab & ac \\ ab & b^{2} & bc \\ ac & bc & c^{2} \end{array}] [\begin{array}{ccc} a^{2} & ab & ac \\ ab & b^{2} & bc \\ ac & bc & c^{2} \end{array}]

= [\begin{array}{ccc} a^{4} + a^{2} b^{2} + a^{2} c^{2} & a^{3} b + {ab}^{3} + {abc}^{2} & a^{3} c + {ab}^{2} c + {ac}^{3} \\ a^{3} b + {ab}^{3} + {abc}^{2} & a^{2} b^{2} + b^{4} + b^{2} c^{2} & a^{2} bc + b^{3} c + {bc}^{3} \\ a^{3} c + {ab}^{2} c + {ac}^{3} & a^{2} bc + b^{3} c + {bc}^{3} & a^{2} c^{2} + b^{2} c^{2} + c^{4} \end{array}]

= [\begin{array}{ccc} a^{2} (a^{2} + b^{2} + c^{2}) & ab (a^{2} + b^{2} + c^{2}) & ac (a^{2} + b^{2} + c^{3}) \\ ab (a^{2} + b^{2} + c^{2}) & b^{2} (a^{2} + b^{2} + c^{2}) & bc (a^{2} + b^{2} + c^{2}) \\ ac (a^{2} + b^{2} + c^{2}) & bc (a^{2} + b^{2} + c^{2}) & c^{2} [a^{2} + b^{2} + c^{2}] \end{array}]

since a² + b² + c² = 1

= [\begin{array}{ccc} a^{2} & ab & ac \\ ab & b^{2} & bc \\ ac & bc & c^{2} \end{array}] = A

⇒ A² = A

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