00:01
Hello everyone, so this is the question that we have part a, that is my matrix 3 minus 2, 1, an eigenvector of a, which is minus 4, 3, 3, 3, 2, 2, negative 3, negative 2, and negative 1, 0, 0 ,000, if he asks, what is the eigen values? but the question is that if we have a matrix b which is 3420 and 000, we need to find the eigenvalues and vectors.
00:53
Now, and we have to verify that the trace equals to the sum of eigenvectors.
01:06
Values and the determinant equals their product.
01:19
Now let's jump onto the solution of the question.
01:22
Now if this is a matrix, part a.
01:25
Now let us say that lambda is the eigenvalue corresponding to the matrix.
01:35
So, determinant of a minus lambda i would be equal to the matrix, which is 3 minus 2 .1.
01:43
So this implies that minus 4 minus lambda 3 3, 2, 3 minus lambda, minus 2, minus 1, 0, minus 2 minus lambda, multiplied by 3 negative 2, 1 is equal to 0, 0 .0.
02:09
Just solve this.
02:11
So this implies this is going to be 3 minus 4 minus lambda, a 6 plus 3 is equal to 0 so 3 lambda is coming out to be negative 15 and lambda hence is coming out to be minus 5 this is my eigenvalue now coming on to part b now it is a proper triple upper triangle matrix right so eigenvalues are going to be the matrix which are 310 so these are my eigen values now for lambda is equal to 3 and we'll use lambda is equal to 1, lambda is equal to 0, and we'll find the subsequent values.
02:53
So this is going to be for lambda is equal to 3.
03:02
3 minus 3, 4, 2, 0, 1, 3, 2, 0, 0 minus 3, 0, 0, then x, y, z is equal to 0, 0.
03:15
0.
03:17
So this is going to be 4y plus 2 z is equal to 0 and z is equal to 0 from here, right? this implies y is also equal to 0.
03:29
Now if you say x is any arbitrary constant, so it will have a matrix.
03:34
My matrix would look something like this, which is x, y, z is equal to k 0, k multiplied by 1 .0.
03:48
Now for lambda is equal to 1.
03:52
So for lambda is equal to 1, i'll have my values as minus 4 z is equal to 0.
03:58
That implies z is equal to 0, 2x plus 4 y is equal to 0, and say, let y is equal to k.
04:05
So this implies x would be equal to minus 2k.
04:08
Again, we formalize this...