 Hello and welcome to the session. I am Deepika here. Let's discuss the question. Solve system of linear equations using matrix method x minus y plus 2 z is equal to 7, 3x plus 4 y minus 5 z is equal to minus 5, 2x minus y plus 3 z is equal to 12. Let's start the solution. Solution, the given system of equations can be written in the form x is equal to b where is equal to 1 minus 1, 2 3 minus 5, 2 minus 1, 3 is equal to x, y, z and b is equal to 7 minus 5, 12 is equal to 1 into 12 minus 5, minus of minus 1 that is plus 1 into 9 minus of minus 10 that is plus 10 and plus 2 into minus 3 minus 8 which is equal to 1 into 7 plus 1 into 19 plus 2 into minus 11 this is equal to 7 plus 19 minus 22. This is again equal to 4 which is not equal to 0. This implies since determinant a is not equal to 0 this implies a is non-singular so inverse exist. Now we will find out a inverse for a inverse. Let us first find out the co-factor of each and every element of a. Now we know that the co-factor of 1 is equal to minus 1 raise to power 1 plus 1 that is i plus j into 12 minus 5 which will be equal to 7 and co-factor of minus 1 is equal to minus 1 raise to power 1 plus 2 because it is the first row and the second column into 9 plus 10 which will be equal to minus 19. So by this way we will find the co-factors of all the elements which are as follows. Now we have found co-factors of all the elements of a. So the matrix formed by the co-factors is 7 minus 19 minus 11 1 minus 1 minus 1 minus 3 11 and 7. Now adjoint a is equal to transpose of this matrix that is the matrix formed by the co-factors. So we get 7 minus 19 minus 11 1 minus 1 minus 1 minus 3, 11 and 7. Now we know that a inverse is equal to 1 over determinant a into adjoint a this is equal to 1 over 4 into 7 minus 19 minus 11 1 minus 1 minus 1 minus 3, 11 and 7. Now a x is equal to b implies a inverse a x is equal to a inverse b which implies i x is equal to a inverse b as a a inverse is equal to i. This implies x is equal to a inverse b this implies x is equal to 1 by 4 into 7 minus 19 minus 11 1 minus 1 minus 1 minus 3, 11 and 7 into b is equal to a 7 minus 5 and 12. Now we will solve this this is equal to 1 by 4 into 7 7s of 49 minus 5 minus 36 now minus 19 into 7 minus 133 plus 5 plus 132 and again this is minus 77 plus 5 plus 84 this will be equal to 1 by 4 into y z so by quitting the corresponding elements we get equal to 2 y is equal to 1 and z is equal to 3 hence we have solved the above system of linear equations using the matrix method and our answer is x is equal to 2 y is equal to 1 and z is equal to 3 I hope the question is clear to you