Using properties of determinants, prove the following :
\begin{vmatrix} a^{2} &bc &ac+c^{2} \\ a^{2}+ab&b^{2} &ac \\ ab& b^{2}+bc & c^{2} \end{vmatrix}= 4a^{2}b^{2}c^{2}

 

 

 

 
 
 
 
 

Answers (1)

\begin{vmatrix} a^{2} &bc &ac+c^{2} \\ a^{2}+ab&b^{2} &ac \\ ab& b^{2}+bc & c^{2} \end{vmatrix}= Let\:\triangle = LHS
taking a,b,c common from c1, cand c3 resp.
\Rightarrow \triangle = abc\begin{vmatrix} a & c &a+c \\ a+b& b & a\\ b &b+c &c \end{vmatrix}
By c_{3}\rightarrow c_{3}-\left ( c_{1}+c_{2} \right )
\Rightarrow \triangle = abc\begin{vmatrix} a & c &0 \\ a+b& b & -2b\\ b &b+c &-2b \end{vmatrix}
By R_{2}\rightarrow R_{2}-R_{3}
\Rightarrow \triangle = abc\begin{vmatrix} a & c &0 \\ a& -c & 0\\ b &b+c &-2b \end{vmatrix}
Expanding along c3 
\Rightarrow \triangle = abc\left \{ 0-0+\left ( -2b \right )\left ( -2ac \right ) \right \}= 4a^{2}b^{2}c^{2}= RHS\cdot



 

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