®
–1
1
A
(
A adjA)
=
® If
a1 x + b1 y + c1 z = d1
a2 x + b2 y + c2 z = d2
a3 x + b3 y + c3 z = d3,
then these equations can be written as A X = B, where
1
1
1
1
2
2
2
2
3
3
3
3
A
,X=
and B=
a
b
c
x
d
a
b
c
y
d
a
b
c
z
d




 




 
=



 




 
 




® Unique solution of equation AX = B is given by X = A–1 B, where A
≠0 ® A system of equation is consistent or inconsistent according as its solution
exists or not ® For a square matrix A in matrix equation AX = B
(i) |A| ≠ 0, there exists unique solution
(ii) |A| = 0 and (adj A) B ≠ 0, then there exists no solution
(iii) |A| = 0 and (adj A) B = 0, then system may or may not be consistent Rationalised 2023-24
DETERMINANTS     103
Historical Note
The Chinese method of representing the coefficients of the unknowns of
several linear equations by using rods on a calculating board naturally led to the
discovery of simple method of elimination