Question

without expanding evaluate the following determinants.
1 a b+c
1 b c+a
1 c a+b​

Answer 1

Step-by-step explanation:

∣

1

1

1

a

b

c

bc

ca

ab

∣

∣

∣

∣

∣

∣

∣

∣

=

abc

1

∣

∣

∣

∣

∣

∣

∣

∣

a

b

c

a

2

b

2

c

2

abc

bca

cab

∣

∣

∣

∣

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

∣

∣

∣

a

b

c

a

2

b

2

c

2

1

1

1

∣

∣

∣

∣

∣

∣

∣

∣

=−

∣

∣

∣

∣

∣

∣

∣

∣

a

2

b

2

c

2

a

b

c

1

1

1

∣

∣

∣

∣

∣

∣

∣

∣

[Interchanging C

1

and C

2

]

=−

∣

∣

∣

∣

∣

∣

∣

∣

a

2

a

1

b

2

b

1

c

2

c

1

∣

∣

∣

∣

∣

∣

∣

∣

[ Taking transpose. Since determinants remains unaltered under transpose]

Comparing this with the given problem we have,

λ=−1.