Inverse matrix

Find out the inverse using Gauss elimination or the reduction method.
A=[2/3. 1
-3. 1/3]

Correct answer:

a = 0.1034
b = -0.3103
c = 0.931
d = 0.2069

Step-by-step explanation:

2/3a+1c = 1
2/3b + 1d = 0
-3a + 1/3c = 0
-3b + 1/3d = 1

2/3·a+1·c = 1
2/3·b + 1·d = 0
-3·a + 1/3·c = 0
-3·b + 1/3·d = 1

2a+3c = 3
2b+3d = 0
9a-c = 0
9b-d = -3

Pivot: Row 1 ↔ Row 3
9a-c = 0
2b+3d = 0
2a+3c = 3
9b-d = -3

Row 3 - 2/9 · Row 1 → Row 3
9a-c = 0
2b+3d = 0
3.2222c = 3
9b-d = -3

Pivot: Row 2 ↔ Row 4
9a-c = 0
9b-d = -3
3.2222c = 3
2b+3d = 0

Row 4 - 2/9 · Row 2 → Row 4
9a-c = 0
9b-d = -3
3.2222c = 3
3.2222d = 0.6667

d = 0.66666667/3.22222222 = 0.20689655
c = 3/3.22222222 = 0.93103448
b = -3+d/9 = -3+0.20689655/9 = -0.31034483
a = 0+c/9 = 0+0.93103448/9 = 0.10344828

a = 3/29 ≈ 0.103448
b = -9/29 ≈ -0.310345
c = 27/29 ≈ 0.931034
d = 6/29 ≈ 0.206897

Our linear equations calculator calculates it.