Solve this
Question:

$\left|\begin{array}{ccc}b^{2}+c^{2} & a b & a c \\ b a & c^{2}+a^{2} & b c \\ c a & c b & a^{2}+b^{2}\end{array}\right|=4 a^{2} b^{2} c^{2}$

Solution:

$\Delta=\left|\begin{array}{ccc}b^{2}+c^{2} & a b & a c \\ b a & c^{2}+a^{2} & b c \\ c a & c b & a^{2}+b^{2}\end{array}\right|$

$\left|\begin{array}{ccc}a\left(b^{2}+c^{2}\right) & a^{2} b & a^{2} c \\ b^{2} a & b\left(c^{2}+a^{2}\right) & b^{2} c \\ c^{2} a & c^{2} b & c\left(a^{2}+b^{2}\right)\end{array}\right| \quad[$ Multiplying the three rows by $a, b$ and $c]$

$=\frac{a b c}{a b c}\left|\begin{array}{ccc}b^{2}+c^{2} & a^{2} & a^{2} \\ b^{2} & c^{2}+a^{2} & b^{2} \\ c^{2} & c^{2} & a^{2}+b^{2}\end{array}\right| \quad$ [Taking out $a, b a n d c$ common from the three columns]

$=\left|\begin{array}{ccc}2\left(b^{2}+c^{2}\right) & 2\left(a^{2}+c^{2}\right) & 2\left(a^{2}+b^{2}\right) \\ b^{2} & c^{2}+a^{2} & b^{2} \\ c^{2} & c^{2} & a^{2}+b^{2}\end{array}\right| \quad$ [Applying $R_{1} \rightarrow R_{1}+R_{2}+R_{3}$ ]

$=2\left|\begin{array}{ccc}b^{2}+c^{2} & a^{2}+c^{2} & a^{2}+b^{2} \\ -c^{2} & 0 & -a^{2} \\ -b^{2} & -a^{2} & 0\end{array}\right|$ [Taking out 2 common from the three columns and then applying $R_{2} \rightarrow R_{2}-R_{1}$ and $\left.R_{3} \rightarrow R_{3}-R_{1}\right]$

$=2\left|\begin{array}{ccc}0 & c^{2} & b^{2} \\ -c^{2} & 0 & -a^{2} \\ -b^{2} & -a^{2} & 0\end{array}\right|$      [Applying $R_{1} \rightarrow R_{1}+R_{2}+R_{3}$ ]

$=2\left\{\left[-c^{2}\left(-a^{2} b^{2}\right)\right]+\left[b^{2}\left(c^{2} a^{2}\right)\right]\right\}$                [Expanding along $R_{1}$ ]

$=4 a^{2} b^{2} c^{2}$

 

Administrator

Leave a comment

Please enter comment.
Please enter your name.