matlab can carry out a wide array (no pun intended) of operations on the matrices that you define.
Try addition and multiplication for starters. Observe what happens:

```
A = [1 2; 3 4]
B = [5 6; 7 8]
AplusB = A + B
AtimesB = A * B
```

**A**times

**B**, which is the result of standard matrix multiplication from linear algebra. However, often one is interested in the element by element multiplication of two matrices. The way to do this in matlab is using the .* operator, where the dot indicated element by element calculation. Try it, and compare the results:

` AtimeseeB = A .* B`

` C=[1 2; 3 4]`

` C'`

` D=[1+i 2; 3 4]`

` D'`

` D.'`

Theorem: If the sizes of the matrices are such that the stated operations can be performed, then:

- \( \left({\bf A}^T \right)^T = {\bf A} \) for any matrix
**A**; - \( \left( {\bf A} + {\bf B} \right)^T = {\bf A}^T + {\bf B}^T .\)
- \( \left( {\bf A} - {\bf B} \right)^T = {\bf A}^T - {\bf B}^T .\)
- \( \left( k\,{\bf A} \right)^T = k\, {\bf A}^T .\)
- \( \left( {\bf A} \, {\bf B}\right)^T = {\bf B}^T \, {\bf A}^T . \)

```
Complete
```