 Hi and welcome to the session. I am Shashik and I am going to help you to solve the following question. Question is, if A is equal to matrix sin alpha cos alpha minus cos alpha sin alpha, then verify A transpose A is equal to I, where I is the identity matrix. Let us start with the solution now. We know A is equal to matrix sin alpha cos alpha minus cos alpha sin alpha. Now, we can get A transpose by interchanging the rows and columns of A. So we get A transpose is equal to sin alpha cos alpha minus cos alpha sin alpha. Now, we have to find A transpose A, both the matrices are square matrices of same order, so their multiplication is defined. So sin alpha minus cos alpha cos alpha sin alpha matrix multiplied by matrix sin alpha cos alpha minus cos alpha sin alpha. Multiplying first row with first column we get sin square alpha plus minus cos alpha multiplied by minus cos alpha. Now, multiplying first row with second column we get sin alpha cos alpha plus minus cos alpha multiplied by sin alpha. And multiplying second row with first column we get cos alpha sin alpha plus sin alpha multiplied by minus cos alpha. And multiplying second row with second column we get cos square alpha plus sin square alpha. Now, we know that cos square alpha is plus sin square alpha is equal to 1. So, we will substitute the value of cos square alpha plus sin square alpha equal to 1 in this matrix. We know matrix A transpose A is further equal to sin square alpha plus cos square alpha sin alpha cos alpha minus cos alpha sin alpha cos alpha sin alpha minus sin alpha cos alpha cos square alpha plus sin square alpha. Now, we can see sin alpha cos alpha minus cos alpha sin alpha would be equal to 0. Similarly, cos alpha sin alpha minus sin alpha cos alpha will also be equal to 0. So, we can write A transpose A is equal to sin square alpha plus cos square alpha is 1. This is equal to 0 this is equal to 0 and cos square alpha plus sin square alpha is again equal to 1. But this matrix is having diagonal element equal to 1. So, it is an identity matrix of the order 2 into 2. So, we can write A transpose A is equal to I as this matrix is equal to I right. Hence, verify this completes the session. Hope you enjoyed the session. Have a nice day.