 Hello and welcome to the session. I am Ashmi and I am going to help you with the following problem. Solve the system of the following equations. We have 2yx plus 3 by y plus 10 by z is equal to 4. 4 by x minus 6 by y is equal to 5 by z is equal to 1. 6 by x plus 9 by y minus 20 by z is equal to 2. Now let us write the solution. The given system of equations may be written as a matrix equation. That is ax is equal to b where x is equal to 1 by x 1 by y 1 by z. a is equal to the given matrix 2 3 10 4 minus 6 5 6 9 minus 20 and b is equal to 4 1 2. Now let us find the determinant of a which is equal to 2 multiplied by 120 minus 45 minus 3 multiplied by minus 80 minus 30 plus 10 multiplied by 36 plus 36 which is equal to 2 multiplied by 75 minus 3 multiplied by minus 110 plus 10 multiplied by 72 which is equal to 150 plus 330 plus 720 which is equal to 1200. Now we can see that this is not equal to 0 therefore a inverse exists. Now let us find the cofactors that is a11 will be equal to minus 1 to the power 1 plus 1 120 minus 45 which is equal to 75. Similarly a12 is equal to minus 1 to the power 1 plus 2 multiplied by minus 80 minus 30 which is equal to 110. Similarly we can find the other cofactors which will be a13 is equal to 72, a21 is equal to 150, a22 is equal to minus 100, a33 is equal to minus 24, a23 is equal to 0, a31 is equal to 75, a32 is equal to 30. Now the matrix formed by these cofactors is equal to 75, 110, 72, 150 minus 100, 0, 75, 30 minus 24. Now therefore a joint of a is equal to transpose of this matrix which is equal to 75, 150, 75, 150, sorry, 110 minus 100, 30, 72, 0 minus 24. Therefore a inverse is equal to 1 by determinant of a multiplied by a joint of a which is equal to 1 by 1200 multiplied by determinant 75, 150, 75, 110 minus 100, 30, 72, 0 minus 200. So this is our required a inverse. Now let us proceed. We know that ax is equal to b. Now free multiplying by a inverse. So we get a inverse ax is equal to a inverse b. Now we know that a inverse a is equal to i so it implies ix is equal to a inverse b as a inverse a is equal to i that is the identity matrix. Hence it implies x is equal to a inverse b. Now we have a inverse and we have b. Let us find the value of the matrix x. So which implies x is equal to a inverse is 1 by 1200 multiplied by determinant 75, 150, 75, 110 minus 100. 30, 72, 0 minus 24 multiplied by b that is 4, 1, 2. Now multiplying this we get 1 by 1200 multiplied by first multiplying this row with this column. So we get 75 multiplied by 4 plus 150 multiplied by 1 plus 75 multiplied by 2. Now multiplying this row with this column so we get 110 multiplied by 4 minus 100 multiplied by 1 plus 30 multiplied by 2. Now multiplying the last row with this column so we get 72 multiplied by 4 plus 0 multiplied by 1 minus 24 multiplied by 2. Now solving it further we get 1 by 1200 multiplied by determinant 300 plus 150 plus 150. Now 440 minus 100 plus 60, 288 plus 0 minus 48 which is equal to 1 by 1200 multiplied by determinant 600, 400, 240. Now we have therefore 1 by x, 1 by y, 1 by z is equal to 600 by 1200, 400 by 1200 and 240 by 1200 which is nothing but equal to 1 by 2. 1 by 3, 1 by 5 which implies 1 by x, 1 by y, 1 by z is equal to 1 by 2, 1 by 3, 1 by 5. Now comparing each term in this column so we get it implies 1 by x is equal to 1 by 2 and 1 by y is equal to 1 by 3 and 1 by z is equal to 1 by 5. Which implies x is equal to 2, y is equal to 3 and z is equal to 5. This is our required answer hence x is equal to 2, y is equal to 3 and z is equal to 5. I hope you understood this problem. Bye and have a nice day.