00:01
Hello students here we have been given a system of linear equations the first equation is 2 w minus x plus 3y plus z this is equal to 0 the second equation is 3 w plus 2x plus 4y minus z plus 4y minus z this is equal to 0 this is our second equation the third equation is 5 w minus 2 x minus 2y minus z minus 2y minus z this is equal to 0 this is our third equation and the fourth and final equation is 2 w plus 3x 2 w plus 3x minus 7y minus 5 z this is equal to 0 so we have to solve this system of linear equations using gaussian elimination method so for gaussian elimination method first of all we'll convert this whole equations into an augmented matrix right so for augmented matrix we'll consider only the coefficients of w x y z in the given equations so first of all let's consider the first equation here the coefficients of w x y z are 2 minus 1 3 1 so our matrix first of the matrix will be 2 minus 1 3 1 right and constant is 0 so i'll put here 0 for constant matrix close now coming to the second equation we have coefficients 3 3 2 4 minus 1 and constant is again 0 3 3 equation the coefficients are 5 minus 2 sorry, write it properly, minus 2 and minus 1.
01:35
The constant again is 0.
01:37
Finally the last and final equation, coefficients are 2, 3, minus 7 and minus 5.
01:46
And constant is again 0.
01:47
So this is our augmented matrix right.
01:49
Now we will transform this matrix into different forms.
01:52
So our basic agenda is to convert the diagonal elements to 1 and the elements lying to the bottom of the diagonal to 0.
01:59
Right so that we can calculate the values of wx y z so first of all what we will do we'll just divide first row by 2 so row 1 sorry yeah first row will divided by 2 so row 1 will become row 1 divided by 2 so what what we will be getting our matrix will look like all rows will be same just this just the first row will be divided by 2 so all the elements will be divided by 2 the first element 2 is divided by 2 it will give us 1 the second element minus 1 divided by 2 2 so it will be minus 1 divided by 2 the second element 3 divided by 2 the 4th element 1 divided by 2 and the last element is 0 so the last element is 0 so i close the matrix sorry i'll draw it properly so although it is not made properly so just give me a second yeah it looks decent now coming to the second row third or 4th row they will be unchanged because you are not changing them as such so 3 2 2 4 and 0 third row 5 minus 2 minus 2 minus 1 and 0 final row 2 3 minus 7 minus 5 and 0 right now in the next step we have to convert these 3 elements to 0 as we have discussed earlier that we have to convert all the elements lying to the bottom of the diagonal to 0 so these 3 elements must be converted into 0 so for that what we will do so just give me a minute i'll divide this page now i'll just simply will replace row 2 from row 2 to row 2 minus 3 row 1 right so what we'll be getting here it is 1 and 1 will be multiplied by 3 so this will come 3 and 3 minus 3 will become 0 right similarly we'll replace row 3 from row 3 to row 3 minus 5 row 1 right similarly we'll change row 4 also row 4 will converted as row 4 mine row 4 will be replaced with row 4 minus 2 row 1 right so our matrix will be first row will be same because we are not changing first row so i'll copy the first row and then we'll calculate step by step the values of row 3 row 4 row 5 so i'll leave some space the first row is as we have already calculated the first row is 1 minus 1 divided by 2 3 divided by 2 so first element is 1 then minus 1 divided by 2 then 3 divided by 2 then 3 divided by and last is again one divided by two constant is zero so i will drive it properly it is zero matrix close now coming to the second row so as we have discussed we'll replace row two from row 2 to three row 2 to 3 row 1 so first element will become 0 because here r2 is 3 and r1 is 1 so 3 1 multiplied by 3 will be 3 3 minus 3 will be 0 so first row will be sorry first element will be 0 so i'll write it here first will be 0 now coming to the second second element here we have r2 as 2 and r 3 is minus 1 divided by 2 so this minus 1 divided by 2 will be multiplied by minus 3 plus sorry 3 divided by 2 this will become minus 1 divided by 2 this minus 1 divided by 2 will be multiplied by this minus 3 divided by 2 and r2 is 2 so 3 divided by 2 plus 2 it will be 3 divided by 2 this will become 7 divided by 2 so our second element will be 7 divided by 2 now coming to the next element i'll replace it is this this was just for rough work coming to the next element here we have 4 r2 is 4 and r3 is 3 divided by 2 so this 3 divided by 2 will be multiplied by minus 3 this will become minus 9 divided by 2 and r2 is already 4 minus 9 divided by 2 so this will become minus 1 divided by 2 right minus 1 divided by 2 now for the next element we have r2 is minus 1 and r1 is 1 divided by 2 so this 1 divided by 2 will be multiplied by minus 3 this will become minus 3 divided by 2 so minus 3 divided by 2 and r2 is already minus 1 so this will become minus 1.
06:15
So this will give us minus 5 divided by 2 right minus 5 divided by 2 and last element the constants are already 0 in both the rows so the last element will be 0 itself so our second row will be 0 7 divided by 2 minus 1 divided by 2 minus 5 by 2 0 so i'll write it here 0 7 divide by 2 sorry i will write it clearly 7 divide by 2 this is 7 next element is 0 7 divided 2 minus 1 divided by 2 minus 5 divided by 2 so here it is minus 1 divided by 2 minus 5 divided by 2 and last element is 0 now coming back to our next row first of all let me replace this already is because this was just for the rough work now coming to the third row we are what we are getting here we are replacing row 3 from row 3 minus 4 row 1 so first element will become 0 because here row 3 is 5 and row 1 is 1 so this one will be multiplied by minus 5 this will become minus 5 minus 5 plus 5 will become 0 so first element will be 0 coming to the second element here row 3 is minus 2 so i'll write it here row 3 is minus 2 and row 1 is 1 divided by 2 so this minus 1 divided by 2 this will become plus 5 divided by 2 so this will give us 1 divided by 2 so here we have 1 divided by 2 the next element in the third element r 3 is minus 2 so r 3 is minus 2 and r1 is 3 divided by 2 so this 3 divided by 2 will be multiplied by minus 5 so this will become minus 15 divided by 2 right minus 15 divided by 2 so this will give us minus 19 divided by 2 so minus 19 divided by 2 now next element we have r3 as minus 1 and r1 is 1 divided by 2 so this 1 divided by 2 will be multiplied by minus 5 so this will become minus 5 divided by 2 and r 3 is already minus 1 right so minus 1 so this will become minus 7 divided by 2 minus 7 divided by 2 and last element is 0 in both r3 and r1 so this will already become 0 so our third row will be 0 1 divided by 2 minus 19 divided by 2 so i'll write it here step by step 0 it is look i forgot it 1 divided by 2 minus 19 divided by 2 so it is 1 divided by 2 minus 19 divided by 2 and the 4th element is minus 7 divided by 2 so minus 7 divided by 2 and 0 right now coming to the coming to the final row first of all we will erase this we have the final row as we have row 4 is row 4 will be row 4 minus 2 row 1 so first element will be 0 itself right 2 minus 2 this will be 0 so 1 element will be 0 2 for the second element for the 2 row 4 is 3 so 3 and row 2 is sorry row 1 row 1 is minus 1 divided by 2 so this minus 1 divided by 2 so this minus 1 by 2 will be multiplied by minus 2 so this will become plus 1 right so this will be 3 plus 1 so next element will be 4 second sorry third element minus 7 row 4 is minus 7 row 4 is minus 7 and row 1 is 3 divided by 2 so row 1 is 3 divided by 2 this 3 divided by 2 will be multiplied by minus 3 so we will be getting minus 10 4th element here minus 5 row 1 is minus 5 and row sorry row 4 is minus 5 and row 1 is 1 divided by 2 so this 1 divided by 2 so this 1 by 2 will be multiplied by minus 2 this will become minus 1 so minus 5 minus 1 this will become minus 6 and the last element is 0 in both r 4 and r1 so this will already become 0 so our elements of this row will be 0 4 10 minus 6 0 so 0 4 minus 10 minus 6 and 0 so here is our next form of the augmented matrix now we have what we have to do from here first of all we'll replace r2 from r3 so let's see what we will get we'll replace r2 with r3 so our basic agenda is still converting the diagonal elements to 1 and the elements lying bottom of the diagonal element to 0 so let's see how we are getting here first second third four elements will be same right sorry first second what am i saying here first and fourth row will be same and r2 and r3 will be interchanged so our next matrix will be 1 minus 1 divide by 2 3 divide by 2 1 divided by 2 and 0 matrix close now coming to the second row in place of second row this third row will come so this will be 0 0 1 divided by 2 minus 19 divided by 2 minus 7 divided by 2 and 0 right in place of 3rd row this second row will come so 0 7 divided by 2 minus 1 divided by 2 minus 5 divided 2 and 0 and in the last row it will be same we are not changing the last row so this will be same minus 6 and 0 next what we will do we have to convert the diagonal element to 1 so this element must be converted to 1 so what we'll do we'll multiply this entire row 2 with 2 so r2 will be multiplied by 2 so this will become 2 r2 so let's do this first all other rows will remain same so i'll copy it here so our first row will be same so let me check here what is first row 1 minus 1 divided by 2 3 divided by 2 so 1 minus 1 divided by 2 3 divided by 2 and the final element is 1 divided by 2 0 1 divided by 2 and 0 matrix close now in place of 2 we are multiplying the whole entire row with 2 so this will become 0 1 minus 19 minus 7 so sorry 0 sorry what am i doing here 0 0 it was 1 1 minus 19 minus 7 so 0 1 minus 19 minus 7 and final element was already 0 so 0 rest of the rows in 3rd row and 4 will remain same so 0 7 divide by 2 minus 1 divided by 2 0 7 divided by 2 minus 5 divided by 2 and minus 5 divided by 2 0 minus 5 divide by 2 0 last row will be 0 4 minus 10 minus 6 6 0 0 0 0.
13:09
4 so i will write it properly 0 4 minus 10 minus 6 and 0 so as per our agenda we have to convert the elements lying to the bottom of the diagonal to 0 so this will be 0 this will be 0 so we have to convert these 2 elements to 0 so for this what we will do so i want you to focus here first of all we'll replace row 3 from row 3 to row 3 minus 7 by 2 row 1 sorry row 2 so actually i'll write it fresh so let's write it here because there is no left space here so row 3 will be replaced from row 3 to row 3 minus 7 divided by 2 row 2 because the row 2 is 1 so this will become 1 and row 3 7 divided by 2 so this will become 0 so this element will come out as 0 similarly our row 4 will be row 4 minus 4 row 1 sorry not row 1 it's row 2 so our next matrix will be let's find out our next matrix 1 will be same second row will be same only 3 and 4th rows are changing so 1 minus 1 divide by 2 3 divide by 2 and 0 this is our first row coming to the second row it is also same 0 1 minus 19 minus 19 and minus 7 this is 0 now coming to the third row first of all first element is 0 for both row 3 and row 2 so this will already be 0 now coming to the second element first of all this will be 0 coming to the second element as you have discussed that row 2 is already 1 and row 3 7 divided by 2 so this will become 7 divided 2 minus 7 divide by 2 so this will give us 0 second element will be 0 now coming to the third element, third element i'll write it here third element for row 3 it is minus 1 divided by 2 minus 1 divided by 2 and for row 2 it is minus 19.
15:25
So minus 19 will be multiplied by minus 7 divided by 2 so this will become plus 133 divided by 2 so plus 133 divided by 2 when we will calculate this this will become 132 divide by 2 which will be 66 so our next element will be 66.
15:43
Now final element fourth element here r3 is minus 5 divided by 2 so minus 5 divided by 2 and r 2 is minus 7 so this minus 7 will be multiplied by minus 7 divided by 2 so this will become 49 divided by 2 so plus 49 divided by 2 now when we will calculate this this will come out as 44 divided by 2 which is 22 so our next element will be 22 and last element is 0 so both it is both 0 for both so this will come out as 0 so i'll raise this because this was just for rough work now coming to the next row here are for the first element is both zero for r4 and r2 so this element will already be zero coming to the second element here r2 is r2 is one sorry r2 is 1 and r4 is 4 so this 4 will be multiplied by minus 4 so this will come 4 minus 4 that is 0 so next element will also be 0 coming to the next element that is third element here r4 is minus 10 so minus 10 and r2 is minus 19 so this minus 19 will be multiplied by minus 4 so this will become plus 76 so plus 76 minus 10 this is 66 so 66 and the last element sorry second last element here my r4 is minus 6 r4 is minus 6 and r2 is minus 7 so this minus 7 will be multiplied by minus 4 this will become plus 28 right plus 28 minus 6 is 22 22 and here it is 0 because it is 0 for all both the elements so i'll erase this this is our next form of the augmented this is our next matrix.
17:19
Now as we can see that our third row and fourth row row becomes same.
17:23
So in the next step what we will do we'll just simply subtract row 3 from row 4.
17:28
So row 4 will become row 4 minus row 3.
17:33
So this whole entire row 4 will become 0 because we are subtracting this from itself because row 3 and row 4 are same right.
17:41
So first first of all all the first second third row will remain same so i'll first of all copy them.
17:47
The first second second third rows will be same so i'll copy them just to remain inconsistent so 1 minus 1 divide by 2 3 divided by 2 so 1 minus 1 divided by 2 3 divided by 2 and the last elements are 1 minus 1 sorry 1 divided by 2 and 0 1 divided by 2 constant is 0 now matrix close coming to the next row we have 0 1 minus 19 minus 7 so 1 minus 19 minus 7 so 1 minus 19 minus 7 19 minus 7 this is again 0.
18:27
Now next element here we have 0602, sorry 66 22, 066 22 and 0...