Product of two Determinants
We can write
Here we have multiplied rows by rows. We can
also multiply rows by columns or columns by rows, or columns by columns.
Note:
If D
= |aij| is a determinant of order n, then the value of the
determinant |Aij|, where Aij is the cofactor of aij,
is Dn-1. This is known as
power cofactor formula. .
Example.1
Prove the following by multiplication of determinants
and power co-factor formula.
= 
Solution
(i) First determinant is equal to ( 2abc)2
(ii) Second determinant is direct
multiplication of determinants in row to row . .
(iii) Third determinant is co-factors of the
first determinant and therefore square of the first.
Example.2
For
all values of A,B,C and P,Q,R, show that
Solution:
The given determinant is equal to
