Question

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 will also rate your answers for you. Thank you kindly!

Answer #1

Let T be the linear transformation from R2 to R2, that rotates a
vector clockwise by 60◦ about the origin, then reﬂects it about the
line y = x, and then reﬂects it about the x-axis.
a) Find the standard matrix of the linear transformation T.
b) Determine if the transformation T is invertible. Give detailed
explanation. If T is invertible, ﬁnd the standard matrix of the
inverse transformation T−1.
Please show all steps clearly so I can follow your...

A population of bacteria, growing according to the Malthusian
model, doubles itself in 10 days. If there are 1000 bacteria
present initially, how long will it take the population to reach
10,000?
Please show all work and steps clearly so I can follow your
logic and learn to solve similar ones myself. I will rate your
answer and provide positive feedback. Thank you kindly!

Solve the following initial/boundary value problem:
∂u(t,x)/∂t = ∂^2u(t,x)/∂x^2 for t>0, 0<x<π,
u(t,0)=u(t,π)=0 for t>0,
u(0,x)=sin^2x for 0≤x≤ π.
if you like, you can use/cite the solution of Fourier sine
series of sin^2(x) on [0,pi] = 1/4-(1/4)cos(2x)
please show all steps and work clearly so I can follow your
logic and learn to solve similar ones myself.

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...

6. Let A = 3 −12 4 −1 0 −2 −1 5 −1 . 1 (a) Find all
eigenvalues of A5 (Note: If λ is an eigenvalue of A, then λ n is an
eigenvalue of A n for any integer n.). (b) Determine whether A is
invertible (Check if the constant term of the characteristic
polynomial χA(λ) is non-zero.). (c) If A is invertible, find (i)
A−1 using the Cayley-Hamilton theorem (ii) All the eigenvalues...

Consider the following. A = −5 12 −2 5 , P = −2 −3 −1 −1 (a)
Verify that A is diagonalizable by computing P−1AP. P−1AP = (b) Use
the result of part (a) and the theorem below to find the
eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A
and B are similar n × n matrices, then they have the same
eigenvalues. (λ1, λ2) =

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

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 +
c1?n ? 1 + ? +
cn
So, on setting ? = 0, we obtain that
det (? A) = cn or det (A)
= (? 1)ncn
det (A) =

Find the inverse matrix of
[ 3 2 3 1 ]
[ 2 1 0 0 ]
[ 3 -5 1 0 ]
[ 4 2 3 1 ]
This is all one 4x4 matrix
Required first step add second row multiplied by (-1) to first
row
try to not use fractions at all to solve this problem

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

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