Special Determinants
Symmetric determinant :
If the elements of a determinant are such that aij
= aji, then the determinant is said to be a Symmetric determinant.
The elements situated at equal distances from the diagonal are equal both in
magnitude and sign.
e.g. 
(remember this
formula for quick calculations)
Skew symmetric determinant
:
If aij = -aji
then the determinant is said to be a Skew symmetric determinant. That is all
the diagonal elements are zero and the elements situated at equal distances
from the diagonal are equal in magnitude but opposite in sign. The value of a
skew symmetric determinant of odd order is zero.
e.g.
Circulant Determinants :
The elements of the rows (or columns) are in
cyclic arrangement.
= -
(a + b + c) ´ {(a - b)2+ (b -
c)2 + (c - a)2}
Other Important
Determinants :
q
q
q
q
Example.1
Prove that 
Solution
D
= 
+ 
By applying C1
→ C1 – C2
and C3 → C3 – C2
in both determinants , we get L.H.S. .
= 
= (a – b)( b – c)
= (a – b)(b – c) { (c – a)+ (c – a) } = 2
(a – b) (b – c) (c – a).
