Section 6.1 Eigenvalues and Eigenvectors: Problem 6 (1...

Section 6.1 Eigenvalues and Eigenvectors: Problem 6 (1 point) The matrix A = [[-2,-2,-8],[-1,-1,-4],[1,1,4]] has two real eigenvalues, one of multiplicity 1 and one of multiplicity 2. Find the eigenvalues and a basis of each eigenspace. ?1 = ______ has multiplicity 1, with a basis of ______ ?2 = ______ has multiplicity 2, with a basis of ______ To enter a basis into WeBWork, place the entries of each vector inside of brackets, and enter a list of these vectors, separated by commas. For instance, if your basis is {[1,2,3],[1,1,1]}, then you would enter [1,2,3],[1,1,1] into the answer blank. Note: You can earn partial credit on this problem.

Transcript

00:01 We are given with the matrix minus 2 minus 2 minus 8 and minus 1 minus 1 minus 4 1 1 and 4 which is the given matrix that we have to find.

00:17 We have to find its eigenvalues and eigenvectors.

00:21 We are easily observed that the last two rows are scalar multiple of each other.

00:25 From here determinant of the matrix is 0.

00:28 If the determinant of matrix is 0 that mean there should be an eigenvalue lambda which is 0.

00:36 Now we will find there is another eigenvalue which is 0 or not.

00:43 Take the 2 by 2 matrix this.

00:45 If this 2 by 2 matrix has ranks 2 or we can say the determinant is 0 that mean there should be another eigenvalue that is 0.

00:56 If we consider this 2 by 2 matrix determinant will be 0.

00:59 If we consider this 2 by 2 matrix determinant is also 0.

01:03 If this 2 by 2 matrix determinant is again 0 and last 2 by 2 matrix determinant is again 0 that mean there should be another eigenvalue lambda which is also 0.

01:14 We have find two eigenvalues.

01:16 One is 0 and other is also 0.

01:19 Now we will find the last third eigenvalue.

01:23 We will check what will be it.

01:26 Now the formula for it is trace of the matrix is equal to the sum of eigenvalues.

01:34 Trace is what? trace is the sum of diagonal elements of the matrix.

01:39 Trace of a.

01:41 What is it? it is negative 2, negative 1 and 4.

01:45 If we add it we will get 1.

01:47 Trace of a is 1 that mean the sum of eigenvalues.

01:51 Eigenvalues are 3 here.

01:52 Two of them are 0.

01:53 So lambda 1 plus 0 plus 0 because we have to find this another value lambda that is unknown yet and this is 0 and this is 0.

02:04 From here lambda 1 is 0.

02:07 Lambda 1 is 1.

02:08 We all get the eigenvalues.

02:11 One eigenvalue is 0 with algebraic multiplicity 2 and another eigenvalue 1 with algebraic multiplicity 1.

02:20 Now we will find the eigenvector corresponding to these eigenvalues.

02:26 For lambda equals to 0.

02:30 For lambda equals to 0 the matrix a minus i.

02:36 Lambda i will be there.

02:38 Lambda is 0.

02:39 So lambda 0 into i is 0.

02:43 So a minus 0 matrix into v vector is equal to 0.

02:48 Put this.

02:48 We will get negative 2, negative 2, negative 8 because this term will be 0 so it will not affect the given matrix a.

02:56 Negative 1, negative 1, negative 4, 1, 1, 4.

03:01 Multiply it with vector.

03:02 Vector is x, y, z is equal to 0.

03:09 Find the equations from here.

03:11 From here we will get negative 2x, negative 2y, negative 8y is equal to 0.

03:18 We can easily observe that negative 2 can be taken out common and x and 2 is also there can be common out.

03:27 We are only getting x plus y plus 4 here is z.

03:35 Last vector was z so 8z.

03:38 4z equals to 0...

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