Question

soumon We have
[
begin{aligned}
text { LHS } &=left|begin{array}{lll}
b+c & c+a & a+b
q+r & r+p & p+q
y+z & z+x & x+y
end{array}right|
&=2left|begin{array}{lll}
a+b+c & c+a & a+b
p+q+r & r+p & p+q
x+y+z & z+x & x+y
end{array}right|
end{aligned}
]
[applying ( C_{1} rightarrowleft(C_{1}+C_{2}+C_{3}right) ) and taking out 2 common from
[
=2left|begin{array}{ccc}
a+b+c & -b & -c
p+q+r & -q & -r
x+y+z & -y & -z
end{array}right| quadleft[C_{2} rightarrowleft(C_{2}-C_{1}right), C_{3} rightarrowleft(C_{3}-C_{1}right)right]
]
( =2(-1)(-1) cdotleft|begin{array}{lll}a+b+c & b & c p+q+r & q & r x+y+z & y & zend{array}right| )
[taking out (-1) common from each one of ( C_{2} ) and ( C_{3} )
[
left.=2left|begin{array}{ccc}
a & b & c
p & q & r
x & y & z
end{array}right|=text { RHS } quad text { [applying } C_{1} rightarrow C_{1}-left(C_{2}+C_{3}right)right]
]
Hence, ( L H S=R H S )

# EXAMPLE 40 Show that b+c cta a+b la b q+r r + p p + 9 = 2 p 9 y +z z+x x+y x y

Solution