Consider the system of equations
a1 x + b1 y + c1 z =  d1
a2 x + b2 y + c2 z = d 2
a3 x + b3 y + c3 z = d 3
Let
A =
1
1
1
1
2
2
2
2
3
3
3
3
, X
and B
a
b
c
x
d
a
b
c
y
d
a
b
c
z
d




 




= 
=




 




 
 




Then, the system of equations can be written as, AX = B, i e ,
 
1
1
1
2
2
2
3
3
3
a
b
c
x
a
b
c
y
a
b
c
z

  

  

  

  
 


 =
1
2
3
d
d
d










Case I If A is a nonsingular matrix, then its inverse exists Now
AX = B
or
A–1 (AX) = A–1 B
(premultiplying by A–1)
or
(A–1A) X = A–1 B
(by associative property)
or
I X = A–1 B
or
X = A–1 B
This matrix equation provides unique solution for the given system of equations as
inverse of a matrix is unique