Publish On: 2019-03-09

# Ben joy

Total Post: 542

## Question: Describe the positive integral powers of matrices

Reply On: 2013-10-07

# Mjay Jollay

Total Post: 0

## ANS: Describe the positive integral powers of matrices

We have seen that the product A.A is defined if and only if A is a square matrix. Let A be n square matrix. We denote the product A.A by A

Also A

so that A

We write A

By the generalized associative law, the product of any finite numbers of matrices is associative and hence we can write down the product A.A.A, ….., A m times equal to A

= A

Similarly, we have

(A

^{2}. Obviously A^{2}is an n × n matrix. Again the product A^{2}.A is defined and is an n × n matrix.Also A

_{2}.A (A.A)A = A.(A.A) = A.A^{2}[By Associative law]so that A

^{2}.A = A.A^{2}= A.A.A = A^{3}.We write A

^{3}= A^{2}.A = A.A^{2}= A.A.A.By the generalized associative law, the product of any finite numbers of matrices is associative and hence we can write down the product A.A.A, ….., A m times equal to A

^{m}.= A

^{m+n}Similarly, we have

(A

^{m})^{n}= A^{mn}.Like Us On Facebook for All Latest Updates