Given system of equations is 3x1+7x2+x3=2 x1+2x2+x3=3 2x1+3x2+4x3=13 The coefficient matrix
A=|
3
7
1
1
2
1
2
3
4
|;|A|=|
3
7
1
1
2
1
2
3
4
|:3(8−3)−7(4−2)+1(3−4)=0 adj(A)=[
5
−2
−1
−25
10
5
5
−2
−1
]T=[
5
−25
5
−2
10
−2
−1
5
−1
] adj(A)B=[
5
−25
5
−2
10
−2
−1
5
−1
][
2
3
13
] =[
10−75+65
−4+30−25
−2+15−13
][
0
0
0
]
Hence, it has infinite number of solutions. Alternate method Augmented matrix [A,B]=[
3
7
1
⋮
2
1
2
1
⋮
3
2
3
4
⋮
13
]
R1↔R2∽[
1
2
1
⋮
3
3
7
1
⋮
3
2
3
4
⋮
13
]
Use operations. R2→R2−3R1,R3→R3−2R1 ∽[
1
2
1
⋮
3
0
1
−2
⋮
−7
0
−1
2
⋮
7
] R3→R2+R3, ∽[
1
2
1
⋮
3
0
1
−2
⋮
−7
0
0
0
⋮
0
] Here, Rank of [A,B]= Rank of A So, the system of equation is consistent. Also. here rank of A< Number of unknowns i.e. 2<3 Hence, the system has infinitely many solutions.