00:01
Hello everyone this question the matrix is given we have to find the eigenvalue and eigenspace for this matrix.
00:11
So for eigenvalue i write this as determinant of minus 1 minus lambda 0 4 minus 1 minus lambda 0 minus 2 0 minus 1 minus lambda equal to 0.
00:26
So evaluating this determinant minus 1 minus lambda into minus 1 minus lambda the whole square another term will become 0.
00:37
So minus 1 minus lambda taking minus common we will be having 1 plus lambda the whole square equal to 0.
00:48
So we will be having minus 1 minus lambda minus into minus will be so 1 plus lambda the whole square equal to 0.
00:55
So this gives you minus 1 minus lambda 1 plus lambda square plus 2 lambda equal to 0.
01:04
So this gives you minus 1 minus lambda square minus 2 lambda minus lambda minus lambda cube minus 2 lambda square equal to 0.
01:15
So this gives you minus lambda cube minus 3 lambda minus 3 lambda square minus 1 equal to 0.
01:23
So this can be rewritten as lambda cube plus 3 lambda plus 3 lambda square plus 1 equal to 0.
01:34
So from this we can found lambda by synthetic division 1 3 3 1.
01:40
So if i take minus 1 i will be having minus 1 2 minus 2 1 minus 1 which is 0.
01:49
So this gives you lambda equal to minus 1 is one root and another root will be lambda square plus 2 lambda plus 1 equal to 0 which can be written as lambda plus lambda plus 1 equal to 0.
02:04
So this gives you lambda plus 1 plus 1 into lambda plus 1.
02:09
So this gives you lambda 1 into lambda 1 equal to 0.
02:13
So this gives the value lambda minus 1 minus 1.
02:16
So we can say that the eigenvalue for this matrix is lambda equal to minus 1.
02:24
So we have to find eigenspace for this lambda.
02:28
Next we have to find the eigenspace for lambda equal to minus 1.
02:36
For this we have to find a minus lambda i that is a is minus 1 0 0 4 minus 1 0 minus 2 0 minus 1...