 Hello friends, welcome to the session on Malka. We are going to discuss matrices. Our given question is if a equal to matrix 3 minus 4 1 minus 1, then prove that a to the power n equal to matrix 1 plus 2 n minus 4 n n 1 minus 2 n, where n is any positive integer. Now let's start with the solution. The basic idea behind the question is mathematical induction method. In this method we first show that the result is true for n equal to 1. Then we assume that the result is true for n equal to k and we prove that it is true for n equal to k plus 1. If the result is true for n equal to k plus 1, then we conclude that the result is true for all natural numbers. We are given a equal to matrix 3 minus 4 1 1 and let this be our first equation. Now we have to show that a to the power n equal to 1 plus 2 n minus 4 n n 1 minus 2 n matrix. Now we will show the result for n equal to 1. a1 equal to 1 plus 2 into 1 minus 4 into 1 1 and 1 minus 2 into 1. This implies a equal to 1 plus 2 minus 4 1 and 1 minus 2. This implies a equal to 3 minus 4 1 and minus 1. Hence the result is true for n equal to 1. Now we assume that the result is true for n equal to k that is a to the power k equal to matrix 1 plus 2 k minus 4 k k and 1 minus 2 k. Now we will show that the result holds for n equal to k plus 1. Now we will substitute the value of a k from equation number third and a from equation number first. This gives us a k plus 1 equal to 1 plus 2 into k minus 4 k k 1 minus 2 k into a which is 3 minus 4 1 and minus 1. This is equal to 1 plus 2 k into 3 plus minus 4 k into 1 then 1 plus 2 k into minus 4 plus minus 4 k into minus 1. Then k into 3 plus 1 minus 2 k into 1 k into minus 4 plus 1 minus 2 k into minus 1. So this is equal to 3 plus 2 k minus 4 minus 4 k k plus 1 and minus 2 k minus 1. This is equal to 1 plus 2 k plus 2 minus 4 common k plus 1 k plus 1 1 minus 2 k minus 2. This is equal to 1 plus 2 common k plus 1 minus 4 k plus 1 k plus 1 and this is 1 minus 2 common k plus 1 and this shows that the result is true for n equal to k plus 1. Therefore by the principle of mathematical inaction the result is true for any natural number n. Hope you understood the solution and enjoyed the session goodbye and take care.