### Matrices: Matrices in MATLAB

### Row reduction and determinant

MATLAB provides row and column operations to compute reduction processes in matrices numerically, but calculate the reduced row echelon form of a matrix with the command `rref`

. This allows you to perform the method of row reduction for calculating an inverse matrix.

Computing the inverse of a matrix

```
>> A = [3 1 -4; -1 0 2; 1 1 -1] % invertible matrix
A =
3 1 -4
-1 0 2
1 1 -1
>> AI = [A eye(3)] % augmented matrix
AI =
3 1 -4 1 0 0
-1 0 2 0 1 0
1 1 -1 0 0 1
>> rref(AI) % reduced row echelon form
ans =
1 0 0 2 3 -2
0 1 0 -1 -1 2
0 0 1 1 2 -1
>> ans(:, 4:6) % submatrix equal to inverse of A
ans =
2 3 -2
-1 -1 2
1 2 -1
>> A^(-1)
ans =
2.0000 3.0000 -2.0000
-1.0000 -1.0000 2.0000
1.0000 2.0000 -1.0000
```

The determinant of the matrix \(A\) in the above example can be calculated with the command `det`

, and the result shows that the matrix is invertible. We give below also an example of a non-invertible matrix because the determinant is equal to 0 and the reduced echelon form is not equal to an identity matrix. If you're still trying to calculate the inverse, you actually get a result with a matching warning. In addition, be always on guard for numerical rounding errors in calculations.

Determinant of a matrix

```
>> A % invertible matrix
A =
3 1 -4
-1 0 2
1 1 -1
>> det(A) % determinant of A
ans =
-1.0000
>> B = [1 -2 -2; 1 14 14; -4 -24 -24] % non-invertible matrix
B =
1 -2 -2
1 14 14
-4 -24 -24
>> det(B) % determinant of B
```

ans =

0

>> rank(B) % rang van B

ans =

2

>> rref(B) % reduced row echelon form of B
ans =
1 0 0
0 1 1
0 0 0
>> B^(-1)
Warning: Matrix is singular to working precision.
> In matlab.internal.math.mpower.viaMtimes (line 35)
ans =
Inf Inf Inf
Inf Inf Inf
Inf Inf Inf