2x – 3y + 5 = 11
3x + 2y – 4z = -5
x + y – 2z= -3
System of equations can be written as AX = B
Where, A = [

2	−3	5
3	2	−4
1	1	−2

]
X = [

x

y

z

] , B = [

11

−5

−3

]
A = [

2	−3	5
3	2	−4
1	1	−2

]
|A| = 2(- 4 + 4) + 3 ( - 6 + 4) + 5 (3 - 2)
|A| = - 6 + 5 = - 1 ≠ 0
∴ A−1 exists and system of equations has a unique solution
A−1 =

1

|A|

(adj A)
adj A = [

0	−1	2
2	−9	23
1	−5	13

]
A−1 =

1

|A|

[

0	−1	2
2	−9	23
1	−5	13

]
= [

0	1	−2
−2	9	−23
−1	5	−13

]
X = A−1B = [

0	1	−2
−2	9	−23
−1	5	−13

] [

11

−5

−3

]
X = [

−5+6

−22−45+69

−11−25+39

] = [

1

2

3

]
So x = 1 , y = 2 , z = 3