Question

Suppose that a 4 × 4 matrix *A* has eigenvalues
*?*_{1} = 1, *?*_{2} = ? 2,
*?*_{3} = 4, and *?*_{4} = ? 4. Use
the following method to find det (*A*).

If

*p*(*?*) = det (*?I* ? *A*) =
*? ^{n}* +

So, on setting *?* = 0, we obtain that

det (? *A*) = *c _{n}* or det (

det (*A*) =

Answer #1

Matrix A is given as A =
0 2 −1
−1 3 −1
−2 4 −1
a) Find all eigenvalues of A.
b) Find a basis for each eigenspace of A.
c) Determine whether A is diagonalizable. If it is, ﬁnd an
invertible matrix P and a diagonal matrix D such that D =
P^−1AP.
Please show all work and steps clearly please so I can follow
your logic and learn to solve similar ones myself. I...

2
1
1
1
0
1
1
1
0
These questions have got me confused:
1. By calculation, I know this matrix has eigenvalue -1, 0, 3
and they are distinct eigenvalues. Can I directly say that this
matrix is diagonalizable without calculating the eigenspace and
eigenvectors? For all situations, If we get n number of answers
from (aλ+b)n , can we directly ensure that the matrix is
diagonalizable?
2. My professor uses CA(x)=det(λI-A) but the textbook
shows CA(x)=det(λI-A). which...

find all eigenvalues and eigenvectors of the given matrix
A= [3 2 2
1 4 1
-2 -4 -1]

Find the eigenvalues and the eigenvectors corresponding to them
of the matrix
-2
1
3
0
-2
6
0
0
4

Find all eigenvectors of this 3x3 matrix, when the eigenvalues
are lambda = 1, 2, 3
4
0
1
-2
1
0
-2
0
1

Normally, we start with a matrix and find the eigenvalues and
eigenvectors. But it’s interesting to see if this process can be
performed in reverse.
Suppose that a 2x2 matrix has eigenvalues of +2 and -1 but no
info on the eigenvectors. Can you find the matrix? How many
matrices would have these eigenvalues?

Normally, we start with a matrix and find the eigenvalues and
eigenvectors. But it’s interesting to see if this process can be
performed in reverse. Suppose that a 2x2 matrix has eigenvalues of
+2 and -1 but no info on the eigenvectors. Can you find the matrix?
How many matrices would have these eigenvalues?

Find all eigenvalues and eigenvectors for the 3x3 matrix
A= 1 3 2
-1 2 1
4 -1 -1

The matrix [−1320−69] has eigenvalues λ1=−1 and
λ2=−3.
Find eigenvectors corresponding to these eigenvalues. v⃗ 1= ⎡⎣⎢⎢
⎤⎦⎥⎥ and v⃗ 2= ⎡⎣⎢⎢ ⎤⎦⎥⎥
Find the solution to the linear system of differential equations
[x′1 x′2]=[−13 20−6 9][x1
x2] satisfying the initial conditions
[x1(0)x2(0)]=[6−9].
x1(t)= ______ x2(t)= _____

find the eigenvalues of the following matrix. then find the
corresponding eigenvector(s) of one ofthose eigenvalues (pick your
favorite).
1 -2 0
-1 1 -1
0 -2 1

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