1. Tardigrade
  2. Question
  3. Mathematics
  4. Matrix A is such that A2=2A-I, where I is the identity matrix, then for n ge 2, An is equal to

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. Matrix $A$ is such that $ {{A}^{2}}=2A-I, $ where $ I $ is the identity matrix, then for $ n\ge 2,\,\,{{A}^{n}} $ is equal to

J & K CETJ & K CET 2012Matrices

Solution:

Given, $ {{A}^{2}}=2A-I $
Now, $ {{A}^{3}}={{A}^{2}}.A $
$ =(2A-I)A $ $ =2{{A}^{2}}-A $
$ =2(2A-I)-A $ $ =3A-2I $
Now, $ {{A}^{4}}={{A}^{3}}.A $
$ =(3A-2I).A $ $ =3{{A}^{2}}-2A $
$ =3(2A-I)-2A $ $ =4A-3I $
Similarly, $ {{A}^{n}}=nA-(n-1)I $