00:01
In the question, the given matrix a is equals to 1 3 1 4 2 7 3 9 1 5 3 1 1 2 0 8.
00:20
So we have to find null column and row space for the given matrix.
00:33
For this we have to apply elementary row operations.
00:36
So, first step operations are r2 r2 -2 r1 because first we have to make these three values as 0 by using this value r3 -r3 gives r3 -r1.
00:56
Similarly r4 is r4 -r1.
01:00
By applying these elementary row operations we get 1 3 1 4 2 -2 is 0 7 -6 is 1 3 -2 is 1 4 -9 -8 is 1 1 -1 is 0 5 -3 is 2 3 -1 is 2.
01:26
Similarly 1 -4 is -3 0 1 -1 4.
01:34
Now in next row operations we have to make these values 0 by using this value.
01:44
So the row operations are r3 gives r3 -2 r2 r4 gives r4 -r2.
01:54
So the resultant reduced matrix is 1 1 3 1 4 0 1 1 1 0 0 0 -5 0 0 0 5.
02:14
Now in third step we have to make this value 0 by using this.
02:19
Since both are 0 we have to make this value 0.
02:27
The row operation is r4 gives r4 plus r3.
02:32
The resultant matrix is 1 3 1 4 0 1 1 1 0 0 0 -5 0 0 0 0.
02:45
So for finding null values, null space we need to find eigenvalues.
02:56
So for finding eigenvalues we have to assume a of x is equal to 0.
03:02
A x equal to 0 means this is an augmented matrix that is 1 3 1 4 2 7 3 9 1 5 3 1 1 2 0 8 with these 0 values.
03:21
So it is called an augmented matrix.
03:24
So from this we have to reduce the matrix again such that we get 1 3 1 4 0 1 1 1 0 0 0 -5 0 0 0 0 all the 0 values.
03:45
So this is the reduced augmented matrix.
03:47
From this matrix we need to find the equations.
03:51
The equations they are by multiplying them with x1 x2 x3 and x4 we get x1 plus 3x2 plus x3 plus 4x4 is equal to 0.
04:10
X2 plus x3 plus x4 is equal to 0.
04:17
Minus 5x4 is equal to 0...