Question

If an n × n matrix A has all the n eigenvalues λ1, λ2, ..., λn, prove that det(A) = λ1 · λ2 · ... · λn.

Answer #1

Given that A is a matrix which has n eigenvalues .

Therefore, we know that the characteristics polynomial of A is

.

Again, given that are roots of the characteristics equation P(t) = 0 and P(t) is polynomial of degree n. So we can write P(t) as follows

.This implies that

Now we substitute t = 0 in the above equation, we have

Hence we can say that

. Hence proved.

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)= _____

Prove: If A is an n × n symmetric matrix all of whose
eigenvalues are nonnegative, then xTAx ≥ 0 for all
nonzero x in the vector space Rn.

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) =

Let X1,X2,..., Xn be independent random variables that are
exponentially distributed with respective parameters λ1,λ2,...,
λn.
Identify the distribution of the minimum V =
min{X1,X2,...,Xn}.

Find the characteristic equation and the eigenvalues (and
corresponding eigenvectors) of the matrix. 0 −3 5 −4 4 −10 0 0
4
(a) the characteristic equation (b) the eigenvalues (Enter your
answers from smallest to largest.) (λ1, λ2, λ3) = the corresponding
eigenvectors x1 = x2 = x3 =

The matrix A=
1
0
0
-1
0
0
1
1
1
3x3 matrix
has two real eigenvalues, one of multiplicity 11 and one of
multiplicity 22. Find the eigenvalues and a basis of each
eigenspace.
λ1 =..........? has multiplicity 1, with a basis of
.............?
λ2 =..........? has multiplicity 2, with a basis of
.............?
Find two eigenvalues and basis.

For these two problems, use the definition of eigenvalues.
(a) An n × n matrix is said to be nilpotent if Ak = O
for some positive integer k. Show that all eigenvalues of a
nilpotent matrix are 0.
(b) An n × n matrix is said to be idempotent if A2 =
A. Show that all eigenvalues of a idempotent matrix are 0, or
1.

Let A be an n×n nonsingular matrix. Denote by adj(A) the
adjugate matrix
of A. Prove:
1) det(adj(A)) = (det(A))
2) adj(adj(A)) = (det(A))n−2A

Let A be a (n × n) matrix. Show that A and AT have
the same characteristic polynomials (and therefore the same
eigenvalues). Hint: For any (n×n) matrix B, we have
det(BT) = det(B). Remark: Note that, however, it is
generally not the case that A and AT have the same
eigenvectors!

Let Z1=0.8X1+0.6X2 and Z2=−0.6X1+0.8X2 be the first and second
principal components with corresponding eigenvalues λ1=3.82 and
λ2=0.18. Then, cov(X1,3Z2)=0.76
true or false?

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