The matrix A = [[2, 0, 0], [-4, 4, -8], [-2, 4, -8]] has...

The matrix A = [[2, 0, 0], [-4, 4, -8], [-2, 4, -8]] has eigenvalues -4, 0, and 2. Find its eigenvectors. The eigenvalue -4 is associated with eigenvector ( , , ). The eigenvalue 0 is associated with eigenvector ( , , ). The eigenvalue 2 is associated with eigenvector ( , , ).

Transcript

00:01 In this question we are given this matrix a which is 2 0 0 minus 4 4 minus 8 and this is minus 2 4 minus 8 and also we are given the eigenvalues for this matrix which is lambda 1 equal minus 4 lambda 2 equal 0 and lambda 3 equal 2.

00:22 We have to find out the corresponding eigenvectors for these eigenvalues.

00:27 Now for finding eigenvectors we solve the equation a minus lambda i into v equal 0 and we solve this equation for this v.

00:36 So let us start with lambda 1 when we have lambda eigenvalue equal minus 4 so a minus minus plus 4 i into v1 equal 0 where v1 is equal to x y z.

00:53 So we have to find out the values of x y z.

00:56 Now a plus 4 i will be equal to 2 plus 4 which is 6 0 0 minus 4 4 plus 4 will be 8 minus 8 minus 2 4 minus 8 plus 4 will be equal to minus 4 and on the right hand side we will have 0 0 0.

01:14 So this is the augmented matrix.

01:18 Now we will reduce it to the row reduced aqueon form using the row operations.

01:23 So r1 tends 1 by 6 times r1.

01:27 So this will be equal to 1 0 0.

01:29 This is 0 and rest of the rows will remain same minus 2 4 minus 4 0.

01:38 Now r1 r2 tends r2 plus 4 times r1 and r3 tends r3 plus 2 times r1.

01:49 So this matrix will be equal to 1 0 0 0.

01:55 This will be 0 0.

01:59 Here we will have 8 minus 8 0.

02:02 This will be 4 minus 4 0.

02:05 Now r2 tends 1 by 8 times r2 and r3 tends 1 by 4 times r3.

02:13 So this matrix will be converted to 1 0 0 0 1 minus 1 0 0 1 minus 1 0.

02:20 So clearly if we do r3 tends r3 minus r2.

02:25 So this matrix will be equal to 1 0 0 0 1 minus 1 0 and 0 0 0 0.

02:36 That means x will be equal to 0 y minus z will be equal to 0 and z will be the free variable.

02:45 Free variable.

02:46 So from here we will get that y is equal to z.

02:49 So that means v1 which is equal to x by z will be equal to 0 z z...

Question

The matrix A = [[2, 0, 0], [-4, 4, -8], [-2, 4, -8]] has eigenvalues -4, 0, and 2. Find its eigenvectors. The eigenvalue -4 is associated with eigenvector ( , , ). The eigenvalue 0 is associated with eigenvector ( , , ). The eigenvalue 2 is associated with eigenvector ( , , ).

Please give Ace some feedback