Question
In: Math
Let A∈Mn(R)"> A ∈ M n ( R ) A∈Mn(R) such that I+A"> I + A I+A is invertible. Suppose that
Let
(a) Show that
(b) Show that
Solutions
Expert Solution
Solution
(a) Show that
then we get
Therefore,
(b) Show that
SUN BUNRA answered 2 years agoRelated Solutions
Let R be a ring and n ∈ N. Let S = Mn(R) be the ring...
Let R be a ring and n ∈ N. Let S = Mn(R) be the ring of n × n
matrices with entries in R.
a) i) Let T be the subset of S consisting of the n × n diagonal
matrices with entries in R (so that T consists of the matrices in S
whose entries off the leading diagonal are zero). Show that T is a
subring of S. We denote the ring T by Dn(R).
ii). Show...
Let A be a diagonalizable n × n matrix and let P be an invertible n...
Let A be a diagonalizable n × n matrix and let P be an
invertible n × n matrix such that B = P−1AP is the diagonal form of
A. Prove that Ak = PBkP−1, where k is a positive integer. Use the
result above to find the indicated power of A. A = 6 0 −4 7 −1 −4 6
0 −4 , A5 A5 =
Let A be a diagonalizable n × n matrix and let P be an invertible n...
Let A be a diagonalizable n × n
matrix and let P be an invertible n × n
matrix such that
B = P−1AP
is the diagonal form of A. Prove that
Ak = PBkP−1,
where k is a positive integer.
Use the result above to find A5
A =
4
0
−4
5
−1
−4
6
0
−6
Let F be a field and R = Mn(F) the ring of n×n matrices with entires...
Let F be a field and R = Mn(F) the ring of n×n matrices with
entires in F. Prove that R has no two sided ideals except (0) and
(1).
. Let T : R n → R m be a linear transformation and A the...
. Let T : R n → R m be a linear transformation and A the
standard matrix of T. (a) Let BN = {~v1, . . . , ~vr} be a basis
for ker(T) (i.e. Null(A)). We extend BN to a basis for R n and
denote it by B = {~v1, . . . , ~vr, ~ur+1, . . . , ~un}. Show the
set BR = {T( r~u +1), . . . , T( ~un)} is a...
Let G be a group of order mn where gcd(m,n)=1 Let a and b be elements...
Let G be a group of order mn where gcd(m,n)=1
Let a and b be elements in G such that o(a)=m and 0(b)=n
Prove that G is cyclic if and only if ab=ba
1. Let T : Mn×n(F) → Mn×n(F) be the transposition map, T(A) = At. Compute the...
1. Let T : Mn×n(F) → Mn×n(F) be the transposition map, T(A) =
At. Compute the characteristic polynomial of T. You may wish to use
the basis of Mn×n(F) consisting of the matrices eij + eji, eij −eji
and eii.
2. Let A = (a b c d) (2 by 2 matrix) and let T :
M2×2(F) → M2×2(F) be defined asT (B) = AB. Represent T as a 4×4
matrix using the ordered basis {e11,e21,e12,e22}, and use this
matrix to...
Let A be a m × n matrix with entries in R. Recall that the row...
Let A be a m × n matrix with entries in R. Recall that the row
rank of A means the dimension of the subspace in RN
spanned by the rows of A (viewed as vectors in Rn), and
the column rank means that of the subspace in Rm spanned
by the columns of A (viewed as vectors in Rm).
(a) Prove that
n = (column rank of A) + dim S,
where the set S is the solution space...
Let A ∈ Mat n×n(R) be a real square matrix. (a) Suppose that A is symmetric,...
Let A ∈ Mat n×n(R) be a real square matrix.
(a) Suppose that A is symmetric, positive semi-definite, and
orthogonal. Prove that A is the identity matrix.
(b) Suppose that A satisfies A = −A^T . Prove that if λ ∈ C is
an eigenvalue of A, then λ¯ = −λ.
From now on, we assume that A is idempotent, i.e. A^2 = A.
(c) Prove that if λ is an eigenvalue of A, then λ is equal to 0...
Let R be a ring and f : M −→ N a morphism of left R-modules. Show that:
Let R be a ring and f : M −→ N a morphism of left R-modules. Show that:
c) K := {m ∈ M | f(m) = 0} satisfies the Universal Property of Kernels.
d) N/f(M) satisfies the Universal Property of Cokernels.
Q2. Show that ZQ :a) contains no minimal Z-submodule
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