Matrix Algebra

Other Operations

Other Operations. Given an $n\times n$ matrix (square matrix)

, we can define the

-th power

$\displaystyle A^k = \underbrace{A \cdots A}_{k} .$

Transpose of Matrix. Given an $m\times n$ matrix , we can define the transpose of . For example,

$\displaystyle \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & ... ...ix} a_{11} & a_{21} \\ a_{12} & a_{22} \\ a_{13} & a_{23} \end{bmatrix}$

In particular,

Properties of transpose. If and are appropriately defined, we have and .

Matlab/Octave. Matlab/Octave can compute the power A^k and the transpose A' of A as follows.

> A^5

> A'

EXAMPLE 4. Let $B = \left[\begin{array}{ccc} -5 & 1 & 0\\ 2 &-3 & 4 \end{array}\right]$ and $C = \left[\begin{array}{cccc} 1 & 1 & 1 & 1 \\ -3 & 5 &-2 & 7 \end{array}\right]$ . Compute and .

EXAMPLE 5. Write the function rmatrix() that will create an $n\times m$ matrix with random entries. Then create a $3\times 3$ matrix with random entries between and .