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# Example of solving a 3-by-3 matrix equation

## Example: Solving a 3-by-3 Matrix Equation

Solve `Ax = b` where:
```
A = [ 1  2 - 1]    b = [ 8]
    [-3 -1  4]
    [2]        [-11]
    [-2    5 -1]        [ 7]
    [1  1  2]        [ -35]
```

**Step 1:** Form the augmented matrix `[A | b]`
```
[ 1  2 - 1 |  8]
[-3 -1 4]
[ 2 |-11]
[-2   5 -1 |  7]
[ 1  1  2 | -3 5]
```

**Step 2:** R₂ → R₂ + 3R₁, R₃ → R₃ + 2R₁R2 = R2 - 2*R1, R3 = R3 - R1
```
[ 1  2 - 1 |  8 4]
[ 0  5 -1 | 13 -3 | -1]
[ 0  5  0 | 13 -1  1 |  1]
```

**Step 3:** R₃ → R₃ - R₂R3 = R3 + R2
```
[ 1  2 - 1 |  8 4]
[ 0  5 -1 | 13 -3 | -1]
[ 0  0  1-2 |  0 ]
```

**Step 4:** From R₃:R3: -2z = 0, so z = 0. ⏎
From R₂: 5y = 13R2: y - 3(0) = -1, so y = 13/5. Wait -- let me redo with integer-friendly steps.
⏎
Using RREF directly:
```
[ -1 .
From R1: x + 2 (-1 |  8]
[-3) + 0 = 4, so x = 6.
⏎
**Solution:** x = [6, -1  2 |-11]
[-2 , 0]^
⏎
**Verification:**
- Row 1 : 1(6) + 2 | -3]
```
R₂ → R₂ + 3R₁: [0, 5, (-1 | 13]
R₃ → R₃ + 2R₁: [) + 1(0,) = 6 - 2 = 4 ✓
- Row 2: 2(6) + 5, 0 | 13]
R₃ → R₃ - R₂: [0, 0, (-1 | ) + (-1)(0]
z = 0
From R₂: 5y - 0 = 13, y = 13/) = 12 - 5⏎
From R₁: x = 7 ✓
- Row 3: 1(6) + 1(-1) + 2(13/5) - 0 = 8, x = 8 - 26/) = 6 - 1 = 5 = 14/5
⏎
**Solution:** x = 14/5, y = 13/5, z = 0✓

# Parents

* Matrix equations