JEE Main & Advanced Mathematics Matrices Positive Integral Powers of a Matrix

Positive Integral Powers of a Matrix

Category : JEE Main & Advanced

The positive integral powers of a matrix A are defined only when A is a square matrix.

 

Also then \[{{A}^{2}}=A.A\], \[{{A}^{3}}=A.A.A={{A}^{2}}A\]. 

 

Also for any positive integers \[m\] and \[n,\]

 

(i) \[{{A}^{m}}{{A}^{n}}={{A}^{m+n}}\] 

 

(ii) \[{{({{A}^{m}})}^{n}}={{A}^{mn}}={{({{A}^{n}})}^{m}}\]

 

(iii) \[{{I}^{n}}=I,{{I}^{m}}=I\]                                                

 

(iv) \[{{A}^{0}}={{I}_{n}}\], where A is a square matrix of order \[n\].

 

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