Page 106 - DMTH201_Basic Mathematics-1

Unit 4: Determinants

Notes
Example

a 2 bc b 2 ca c 2 ab a b c 2
Prove that c 2 ab a 2 bc b 2 ca c a b
b 2 ca c 2 ab a 2 bc b c a

Solution:

a b c
Let A, B and C be the cofactors of a, b and c respectively in c a b . We note that the
b c a
determinant on the L.H.S. of the given equation is a determinant of cofactors.

a 2 bc b 2 ca c 2 ab A B C
Let = c 2 ab a 2 bc b 2 ca C A B
1
b 2 ca c 2 ab a 2 bc B C A

A B C a b c
Then = C A B c a b
1
B C A b c a

aA bB cC cA aB bC bA cB aC 0 0
= aC bA cB cC aA bB bC cA aB 0 0
aB bC cA cB aC bA bB cC aA 0 0

3
2
Thus, = or = . Hence the result.
1 1
Note: The solution of the above example is based on property (7) of determinants.

Example

2bc a 2 c 2 b 2 a b c 2
Prove that c 2 2ac b 2 a 2 b c a .
b 2 a 2 2ab c 2 c a b

Solution:

2
a b c a b c a b c
We can write b c a = b c a b c a
c a b c a b c a b

a b c a c b 2bc a 2 c 2 b 2
= b c a b a c = c 2 2ac b 2 a 2
c a b c b a b 2 a 2 2ab c 2

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