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Evaluation of a Determinant

If A=[ a2 ab ...

Question

If $A = ⎡ ⎢ ⎣ \begin{matrix} a^{2} & a b & a c a b & b^{2} & b c a c & b c & c^{2} \end{matrix} ⎤ ⎥ ⎦ and a^{2} + b^{2} + c^{2} = 1,$ then $A^{2} =$

A

A

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B

2 A

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C

3 A

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D

4 A

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Solution

The correct option is A $A$
$A^{2} = ⎡ ⎢ ⎣ \begin{matrix} a^{2} & a b & a c a b & b^{2} & b c a c & b c & c^{2} \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} a^{2} & a b & a c a b & b^{2} & b c a c & b c & c^{2} \end{matrix} ⎤ ⎥ ⎦$

$= ⎡ ⎢ ⎣ \begin{matrix} a^{4} + a^{2} b^{2} + a^{2} c^{2} & a^{3} b + a b^{3} + a b c^{2} & a^{3} c + a b^{2} c + a c^{3} a^{3} b + a b^{3} + a b c^{2} & a^{2} b^{2} + b^{4} + b^{2} c^{2} & a^{2} b c + b^{3} c + b c^{3} a^{3} c + b^{2} a c + a c^{3} & a^{2} b c + b^{3} c + b c^{3} & a^{2} c^{2} + b^{2} c^{2} + c^{4} \end{matrix} ⎤ ⎥ ⎦$
$= ⎡ ⎢ ⎢ ⎣ \begin{matrix} a^{2} (a^{2} + b^{2} + c^{2}) & a b (a^{2} + b^{2} + c^{2}) & a c (a^{2} + b^{2} + c^{2}) a b (a^{2} + b^{2} + c^{2}) & b^{2} (a^{2} + b^{2} + c^{2}) & b c (a^{2} + b^{2} + c^{2}) a c (a^{2} + b^{2} + c^{2}) & b c (a^{2} + b^{2} + c^{2}) & c^{2} (a^{2} + b^{2} + c^{2}) \end{matrix} ⎤ ⎥ ⎥ ⎦$
$= ⎡ ⎢ ⎣ \begin{matrix} a^{2} & a b & a c a b & b^{2} & b c a c & b c & c^{2} \end{matrix} ⎤ ⎥ ⎦ = A (∵ a^{2} + b^{2} + c^{2} = 1)$

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Similar questions

A = ⎡ ⎢ ⎣ \begin{matrix} a^{2} & a b & a c a b & b^{2} & b c a c & b c & c^{2} \end{matrix} ⎤ ⎥ ⎦ and a^{2} + b^{2} + c^{2} = 1,

A^{2} =

A = ⎡ ⎢ ⎣ \begin{matrix} a^{2} & a b & a c a b & b^{2} & b c a c & b c & c^{2} \end{matrix} ⎤ ⎥ ⎦

a^{2} + b^{2} + c^{2} = 1

A^{2} =

A = ⎡ ⎢ ⎣ \begin{matrix} a^{2} & a b & a c a b & b^{2} & b c a c & b c & c^{2} \end{matrix} ⎤ ⎥ ⎦

a^{2} + b^{2} + c^{2} = 1,

A^{2} =

A = ⎡ ⎢ ⎣ \begin{matrix} 0 & c & - b - c & 0 & a b & - a & 0 \end{matrix} ⎤ ⎥ ⎦

B = ⎡ ⎢ ⎣ \begin{matrix} a^{2} & a b & a c a b & b^{2} & b c a c & b c & c^{2} \end{matrix} ⎤ ⎥ ⎦,

A B =

a + b + c = 0

\frac{1}{b^{2} + c^{2} - a^{2}} + \frac{1}{c^{2} + a^{2} - b^{2}} + \frac{1}{a^{2} + b^{2} - c^{2}}

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Evaluation of a Determinant

Standard XII Mathematics

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