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Power Of Matrices

The Matrices can be multiplied to get product matrix and also they demonstrate all other mathematical properties. The power of matrices is another mathematical property of matrix where matrix is raised to a power using an exponent. This brings another question, does the exponent laws applies to matrices or not ? what type of matrices qualifies to be raised to some power ? What about common mathematical identities that involve matrices and power of matrices.

Exponents or Power of a Number

Exponent or power is a number which tell us how many times a number base should multiplied by itself. If a represents a base and n is its power, then its written as a^n which means

Figure 1 - a is raised to power n which means 'a' multiplied n times to get a final product
Figure 1 – a is raised to power n which means ‘a’ multiplied n times to get a final product

Similarly, a Square Matrix A and an integer n is given, then n^th power of A is defined as product matrix obtained by multiplying A by itself n times.

A^n = A \times A \times A \times ... \times A (n times)

Note that the matrix A is

  • a square matrix
  • and A^n is a product matrix of same order.

The exponents have their own algebra which is given as follows.

Basic Laws of Exponents

The basic laws of exponents applied to any real number a \in R and these are

  • a^p \times a^q = a^{p + q}
  • \frac{a^p } {a^q } = a^{p - q}
  • (a^p)^q = a^{p * q}
  • (ab)^q = a^{q} * b^{q}

We need to find out whether these laws applies to square matrices or not. Let us verify this claim with examples.

Example #1

Suppose A is a square matrix of order 2 x 2.

A = \begin{bmatrix}1 & 2\\2 & 3\end{bmatrix}A^2 = \begin{bmatrix}1 & 2\\2 & 3\end{bmatrix} \times \begin{bmatrix}1 & 2\\2 & 3\end{bmatrix} = \begin{bmatrix}1+4 & 2+6\\2+6 & 4+9\end{bmatrix} = \begin{bmatrix}5 & 8\\8 & 13\end{bmatrix}A^3 = \begin{bmatrix}5 & 8\\8 & 13\end{bmatrix} \times \begin{bmatrix}1 & 2\\2 & 3\end{bmatrix} = \begin{bmatrix}5+16 & 10+24\\8+26 & 16+39\end{bmatrix} = \begin{bmatrix}21 & 34\\34 & 55\end{bmatrix}A^2 \times A^3 = A^{2 + 3} = A^{5}A^2 \times A^3 = \begin{bmatrix}5 & 8\\8 & 13\end{bmatrix} \times \begin{bmatrix}21 & 34\\34 & 55\end{bmatrix} = \begin{bmatrix}105+272 & 170+440\\168+442 & 272+715\end{bmatrix} = \begin{bmatrix}377 & 610\\610 & 987\end{bmatrix}

Also,

A^5 = \begin{bmatrix}377 & 610\\610 & 987\end{bmatrix}

Therefore, both side of the equation is equal.

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Power Of Matrices

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