00:01
Here let us assume that a equals to 2 0 1 0 2 minus 1 minus 1 minus 2 2.
00:14
So, the characteristic equation will be given by determinant of a minus lambda i and that is equal to 0.
00:30
So, putting the values here we will get determinant of a that is 2 0 1 0 2 minus 1 minus 1 minus 2 2 minus 1 1 1 lambda 1 1 1 this is identity matrix and that is equals to 0.
00:57
On further solving this we will get determinant of 2 minus lambda 0 minus 1 0 2 minus lambda minus 2 1 minus 1 2 minus lambda and this will be equal to 0.
01:20
On taking determinant of this matrix we will get 2 minus lambda into 2 minus lambda 2 minus lambda minus 2 minus 0 minus 1 0 minus 1 2 minus lambda and that is equals to 0.
01:45
On further simplifying this we will get 2 minus lambda 2 minus lambda square minus 2 minus 1 minus 1 2 minus lambda equal to 0.
02:00
On further solving this we will get the equation 2 minus lambda lambda square minus 4 lambda plus 2 plus 2 minus lambda and that is equals to 0.
02:17
We will get the equation delta cube minus 6 lambda square plus 11 lambda minus 6 and that is equals to 0.
02:28
On solving this cubic equation we will get three values of lambda that is 3 1 and 2.
02:36
So, these are the eigenvalues of this matrix.
02:40
Further for eigenvector, eigenvector for the first eigenvalue that is lambda is equals to 3 we will have a minus 3i v equal to 0 where v is the eigenvector represented as v1 v2 v3.
03:11
On solving this we will get minus 1 0 1 0 minus 1 minus 1 minus 1 minus 2 minus 1 into v1 v2 v3 and that is equals to 0.
03:32
On forming the equations here we will get minus v1 minus v3 equals to 0 minus v2 minus 2 v3 equals to 0 and v1 minus v2 minus v3 is equals to 0.
03:49
So, now let us assume that v3 is equals to t where t is some constant.
03:56
So, value of v1 will be minus t and value of v2 will be minus 2t.
04:03
Here we will get the eigenvector as minus t minus 2t and t.
04:14
On taking t common here we will get minus 1 minus 2 and 1.
04:23
This is eigenvector for eigenvalue lambda is equals to 3 and 1 part of the answer.
04:31
Next eigenvalue eigenvector for lambda is equals to 1...