Question

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).

Answer #1

Let A be an nxn matrix. Prove that A is
invertible if and only if rank(A) = n.

Let A be an invertible matrix. Show that A∗ is invertible, and
that (A∗ ) −1 = (A−1 ) ∗ .

Consider an invertible n × n matrix A. Can you write A = RQ, where
R is an upper triangular matrix and Q is orthogonal?

True or False
(5). Suppose the matrix A and B are both invertible, then (A +
B)−1 = A−1 + B−1
. (6). The linear system ATAx = ATb is always consistent for any
A ∈ Rm×n, b ∈Rm .
(7). For any matrix A ∈Rm×n , it satisﬁes dim(Nul(A)) =
n−rank(A).
(8). The two linear systems Ax = 0 and ATAx = 0 have the same
solution set.
(9). Suppose Q ∈Rn×n is an orthogonal matrix, then the row...

(Linear Algebra)
A n×n-matrix is nilpotent if there is a "r" such that
Ar 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.

Let A m×n be a given matrix with m > n. If the time taken to
compute the determinant of a square matrix of size j is j to
the power 3, find upper bound on the
a) total time taken to find the rank of A using determinants
b) number of additions and multiplications required to determine
the rank using the elimination procedure.

Suppose A ∈ Mm×n(R) is a matrix with rank m. Show that there is
an n × m matrix B such that AB = Im. (Hint: Try to determine
columns of B one by one

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, B ? Mn×n be invertible matrices. Prove the following
statement: Matrix A is similar to B if and only if there exist
matrices X, Y ? Mn×n so that A = XY and B = Y X.

Let A,B be a m by n matrix, Prove that
|rank(A)-rank(B)|<=rank(A-B)

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