 Hello friends, welcome to the session. I am Malkha. We are going to discuss matrices. Our given question is if a equal to matrix 0 minus 10 alpha by 2, 10 alpha by 2 is 0 and i is the identity matrix of order 2, then show that i plus a equal to i minus a into matrix cos alpha minus sin alpha, sin alpha and cos alpha. So, let's start with the solution. We are given a equal to matrix 0 minus 10 alpha by 2, 10 alpha by 2 and 0 is given in the question that i is the identity matrix of order 2. Therefore, i equal to 1, 0, 0, 1. Now, we have to show that i plus a equal to i minus a into matrix cos alpha minus sin alpha, sin alpha and cos alpha. So, we will start with the LHS of the given equation that is i which is equal to i is 0, 0, 1 plus a which is 0 minus 10 alpha by 2, 10 alpha by 2, 0. Now, we will add all these matrices and this implies 1 minus 10 alpha by 2, 10 alpha by 2 and 1 which is LHS. Now, we will go for RHS. We are given that for RHS we have to show that i minus a into matrix cos alpha minus sin alpha, sin alpha cos alpha. This is equal to as we know that i is 1, 0, 0, 1 minus a which is 0 minus 10 alpha by 2, 10 alpha by 2 and 0, then matrix cos alpha minus sin alpha, sin alpha and cos alpha. This is equal to 1 minus 0 is 1, 0 minus of minus 10 alpha by 2 is 10 alpha by 2, then 0 minus 10 alpha by 2 is minus 10 alpha by 2 and 1 minus 0 is 1 into matrix cos alpha minus sin alpha, sin alpha and cos alpha. This is equal to 1 into cos alpha plus 10 alpha by 2 into sin alpha, then 1 into minus sin alpha plus 10 alpha by 2 into cos alpha, then minus 10 alpha by 2 into cos alpha plus 1 into sin alpha and minus 10 alpha by 2 into minus sin alpha plus 1 into cos alpha. This is equal to matrix cos alpha plus 10 alpha by 2 and sin alpha minus sin alpha plus 10 alpha by 2 cos alpha minus 10 alpha by 2 cos alpha plus sin alpha plus 10 alpha by 2 sin alpha plus cos alpha. So, now by using the formula that is cos alpha equal to 1 minus 10 square alpha by 2 upon 1 plus 10 square alpha by 2 and sin alpha equal to 2 10 alpha by 2 upon 1 plus 10 square alpha by 2. Now, we use these two formulas in the above part that is that will be equal to in place of cos alpha we will write 1 minus 10 square alpha by 2 upon 1 plus 10 square alpha by 2 plus 10 alpha by 2 into sin alpha which is 2 10 alpha by 2 upon 1 plus 10 square alpha by 2. Now, minus sin alpha is minus 2 10 alpha by 2 upon 1 plus 10 square alpha by 2 plus 10 alpha by 2 into cos alpha which is 1 minus 10 square alpha by 2 upon 1 plus 10 square alpha by 2 we solve this second row that is minus 10 alpha by 2 into cos alpha which is 1 minus 10 square alpha by 2 upon 1 plus 10 square alpha by 2 plus sin alpha which is 2 10 alpha by 2 upon 1 plus 10 square alpha by 2. Similarly, 10 alpha by 2 into sin alpha which is 2 10 alpha by 2 upon 1 plus 10 square alpha by 2 plus cos alpha which is 1 minus 10 square alpha by 2 upon 1 plus 10 square alpha by 2 this is equal to 1 plus 10 square alpha by 2 and in the numerator we will write 1 minus 10 square alpha by 2 plus 2 10 square alpha by 2 and here we will write minus 2 10 alpha by 2 plus 10 alpha by 2 minus 10 cube alpha by 2 upon 1 plus 10 square alpha by 2 and for the second row minus 10 alpha by 2 plus 10 cube alpha by 2 plus 2 10 alpha by 2 upon 1 plus 10 square alpha by 2 and here 2 10 square alpha by 2 plus 1 minus 10 square alpha by 2 upon 1 plus 10 square alpha by 2 so this is equal to 1 plus 10 square alpha by 2 upon 1 plus 10 square alpha by 2 and here we get minus 10 alpha by 2 minus 10 cube alpha by 2 upon 1 plus 10 square alpha by 2 similarly here we get 10 cube alpha by 2 plus 10 alpha by 2 upon 1 plus 10 square alpha by 2 and here we get 1 plus 10 square alpha by 2 upon 1 plus 10 square alpha by 2 this is equal to 1 minus 10 alpha by 2 common we get 1 plus 10 square alpha by 2 and in the denominator we get 1 plus 10 square alpha by 2 here we take 10 alpha by 2 common we get 10 square alpha by 2 plus 1 upon 1 plus 10 square alpha by 2 and 1 this is equal to 1 minus 10 alpha by 2 10 alpha by 2 and 1 LHS is equal to RHS so it is proved hope you understood the solution and enjoyed the session goodbye and take care