 Hi and welcome to the session. Let us discuss the following question. The question says, using matrices all the following system of linear equations x plus y plus z equals to 4 to x minus y plus z equals to minus 1 to x plus y minus 3 z equals to minus 9. Now begin with the solution. Given equations can be written as books to be where a is equal to 3 by 3 matrix in which elements are coefficient of x, y and z. Equal to column matrix in which elements are x, y, z and b is also equal to column matrix in which elements are 4 minus 1 minus 9. Equals to b implies x is equal to a inverse b. Now let us first find determinant of a to 1 into this determinant minus 1 into determinant of 2, 1, 2 minus 3 plus 1 into determinant of 2 minus 1 to 1. Now this is equal to 1 into 3 minus 1 minus 1 into minus 6 minus 2 plus 1 into 2 plus 2. Now this is equal to plus 4 and this is equal to 40 and this is not equal to 0 so as determinant of a is not equal to 0 so this implies a inverse exists. It is a non-singular matrix a inverse exists. Now find the co-factors of matrix a. Now a11 is equal to minus 1 to the power of 1 plus 1 into determinant of minus 1, 1, 1 minus 3 and this is equal to 2. a12 is equal to minus 1 to the power 1 plus 2 into determinant of 2, 1, 2 minus 3 and this is equal to 8. a13 is equal to minus 1 to the power 1 plus 3 into determinant of 2 minus 1 to 1 and this is equal to 4. Proceeding on the same way we have found out other co-factors also. Now in joint a is equal to transpose of 3 by 3 matrix in which elements are 2, 8, 4, 4 minus 5, 1, 2, 1 minus 3. and this is equal to 3 by 3 matrix in which elements are 2, 4, 2, 8 minus 5, 1, 4, 1 minus 3. a inverse is equal to 1 by determinant of a into a joint a. determinant of a is 40 so we have 1 by 40 into a joint a. this is equal to a inverse b. this is equal to 1 by 40 into this matrix. what you call a matrix in which elements are 4 minus 1 minus 9. and this is equal to 1 by 40 into 3 by 3 matrix in which elements are 2 into 4 plus 4 into minus 1 plus 2 into minus 9. 8 into 4 plus minus 5 into minus 1 plus 1 into 4, 4 into 4 plus 1 into minus 1 plus minus 3 into minus 9. and this is equal to 1 by 14 into column matrix in which elements are minus 14, 28 and 42. this is equal to column matrix in which elements are minus 1, 2, 3. column matrix x, y, z is equal to this column matrix and this implies x is equal to minus 1, y is equal to 2 and z is equal to 3. hence our required answer is x equals to minus 1, y equals to 2 and z equals to 3. this completes the session. Bye and take care.