Question

(Linear Algebra)

A n×n-matrix is nilpotent if there is a "r" such that
A^{r} is the nulmatrix.

1. show an example of a non-trivial, nilpotent 2×2-matrix

2.let A be an invertible n×n-matrix. show that A is not nilpotent.

Answer #1

linear algebra
Chapter based on invertible matrix.
For square matrix A, A is invertible if and only if AT
is
invertible.
Is this statement true/ false. please justify?
thank you

Linear Algebra question:Suppose A and B are invertible
matrices,with A being m*m and B n*n.For any m*n matrix C and any
n*m matrix D,show that:
a)(A+CBD)-1-A-1C(B-1+
DA-1C)-1DA-1
b) If A,B and A+B are all m*m invertible matrices,then deduce
from a) above that
(A+B)-1=A-1-A-1(B-1+A-1)-1A-1

Let A be an m × n matrix, and Q be an n × n invertible
matrix.
(1) Show that R(A) = R(AQ), and use this result to show that
rank(AQ) = rank(A);
(2) Show that rank(AQ) = rank(A).

Let
A be an n by n matrix. Prove that if the linear transformation L_A
from F^n to F^n defined by L_A(v)=Av is invertible then A is
invertible.

Linear Algebra
Find the 2x2 matrix A of a linear transformation T: R^2->R^2
such that T(vi) = wi for i = 1,2.
v1=(4,3), v2=(5,4); w1=(-3,2), w2=(-3,-4)

1.Let A be an n x n matrix. Which of these conditions
show that A is invertible?
•det A= 0
• dim (NulA) = 1
•ColA=R^n
•A^T is invertible
•an n x n matrix, D, exists where AD=In

Question for linear algebra ----diagonalization
If a nxn matrix A is diagonalizable, will their power matrix
AK be diagonalizable?
If there are two non-diagonalizable matrix A and B, will their
product AB must also non-diagonalizable?

n x n matrix A, where n >= 3. Select 3 statements from the
invertible matrix theorem below and show that all 3 statements are
true or false. Make sure to clearly explain and justify your
work.
A=
-1 , 7, 9
7 , 7, 10
-3, -6, -4
The equation A has only the trivial solution.
5. The columns of A form a linearly independent set.
6. The linear transformation x → Ax is one-to-one.
7. The equation Ax...

Prove that for a square n ×n matrix A, Ax = b (1) has one and
only one solution if and only if A is invertible; i.e., that there
exists a matrix n ×n matrix B such that AB = I = B A.
NOTE 01: The statement or theorem is of the form P iff Q, where
P is the statement “Equation (1) has a unique solution” and Q is
the statement “The matrix A is invertible”. This means...

a)Assume that you are given a matrix A = [aij ] ∈ R n×n with (1
≤ i, j ≤ n) and having the following interesting property:
ai1 + ai2 + ..... + ain = 0 for each i = 1, 2, ...., n
Based on this information, prove that rank(A) < n.
b) Let A ∈ R m×n be a matrix of rank r. Suppose there are right
hand sides b for which Ax = b has no solution,...

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