This means, a system of linear equations like
a1 x + b1 y = c1
a2 x + b2 y = c2
can be represented as 
1
1
1
2
2
2
a
b
c
x
a
b
c
y




  =




 
 



 Now, this
system of equations has a unique solution or not, is
determined by the number a1 b2 – a2 b1 (Recall that if
1
1
2
2
a
b
a
≠b
 or, a1 b2 – a2 b1 ≠ 0,  then the system of linear
equations has a unique solution) The number a1 b2 – a2 b1
which determines uniqueness of solution is associated with the matrix 
1
1
2
2
A
a
b
a
b


= 



and is called the determinant of A or det A