00:01
These are the given system of equations that is minus w plus 2x minus 3y plus z is equal to minus 8, minus w plus x plus y minus z is equal to minus 4, w plus x plus y plus z is equal to 22, minus w plus x minus x minus y minus z is equal to minus 40.
00:25
So we have to find the solution of these system of equations by using gauze elevates.
00:31
Method so first let us write down the augmented matrix so this is the augmented matrix of the given system of equations so in the goss elimination method our main objective is to make these diagonal elements one and these elements zero so let us by using row operations so let us start by making the elements in the first column 0 that is this this and this so as we can see that we can make the second row element 0 by subtracting it with the first row in the same way if we take the third row element we can subtract we can add it with the first row to make it zero in the same way the fourth row if we subtract it with the first row it will become zero so the row transformations are r2 tends to here r2 tends to is nothing but it is happening in the second row.
01:45
So r2 tends to r2 minus r1.
01:53
Next is r3 tends to r3 plus r1.
02:03
The same way r4 tends to r4 minus r1.
02:11
So after we use this, this is a row.
02:15
Transformations the resultant is the resultant of the matrix is this so we can see that the first column elements that is the second row element third row element and the fourth row element have become zero and the rest of the elements have changed because we are multiplying the whole first row and adding it with the whole of the second or third or fourth row so the rest of the elements change so our remaining objective is to make this element and this element zero so we can uh in the second column we can make the third row element zero by adding it with the first row which is multiplied by three in the same way in the fourth row we can make it zero by uh subtracting the second row so the row column transformation the row transformations would be r3 tends to r3 plus three are two in the same way r4 tends to r4 minus r1, sorry, r2.
03:31
So the resultant of this matrix after the row column transformation is this one.
03:41
So we can see that in the second column the last two elements have become zero.
03:46
So our remaining objective is to make this element zero.
03:51
So to do that we have to use the row transformation.
03:58
That is we have to multiply the fourth row with five and add it to this third row.
04:05
So the row column transformation would be r4 tends to 5 r4 plus r3.
04:27
So after this row transformation the resultant matrix would be.
04:34
So we can see that we have made the bottom elements that is these elements zero.
04:43
So our remaining objective is that we have to make the diagonal elements one.
04:48
So we can see that in the first row, the diagonal element is minus one.
04:52
In the second row it is minus one.
04:53
In the third row it is 10.
04:55
In the fourth row it is minus four.
04:56
So in the first row we have to multiply the whole we have to divide the whole row by minus 1 in the second in the same way in the second row we have to divide the whole row by minus 1 in the third row we have to divide the whole row by minus 10 in the fourth row divide by minus 4 so the row transformations would be r1 tens to r1 by minus 1 r2 tends to r2 by minus 1 minus 1, r3 tends to r3 by r3 by 10 and r4 tends to r4 by minus 4.
06:02
So the resultant matrix after which we apply after applying these road transformations are...