Some Properties of Matrix Powers

Some properties of multiplication of a matrix by itself to form powers:

Suppose $\mathbf{A},\mathbf{B}$ are $n \times n$ matrices and $p,q$ are positive integers.

  1. Matrix power:
    \begin{displaymath}
  2. Zero power: $\mathbf{A}^0 = \mathbf{I}$
  3. Addition of powers: $\mathbf{A}^p \mathbf{A}^q = \mathbf{A}^{p+q}$
  4. Multiplication of powers: $(\mathbf{A}^p)^q = \mathbf{A}^{pq}$
  5. Distribution of powers: If $\mathbf{A}\mathbf{B} = \mathbf{B}\mathbf{A}$, then $(\mathbf{A}\mathbf{B})^p = \mathbf{A}^p \mathbf{B}^p$.

It does not matter in which order you multiply a matrix by itself to calculate one of its powers.

See the spreadsheet screen shot, below, showing various ways of calculating the fourth power of a matrix A.

Note that the third line shows that all methods of calculating the power yield the same result:

Download Microsoft Excel 2002 (2000-compatible) spreadsheet with the above examples: click here.

Copyright © 2002-2003 Crystal Sloan