Question

Prove:

Why must every upper triangular matrix with no zero entries on the main diagonal be nonsingular?

(Linear Algebra)

Answer #1

A triangular matrix is called unit triangular if
it
is square and every main diagonal element is a 1.
(a) If A can be carried by the gaussian algorithm
to
row-echelon form using no row interchanges,
show that A = LU where L is unit lower
triangular and U is upper triangular.
(b) Show that the factorization in (a) is
unique.

Show that if A is an (n × n) upper triangular matrix or lower
triangular matrix, its eigenvalues are the entries on its main
diagonal. (You may limit yourself to the (3 × 3) case.)

Linear Algebra: Show that the set of all 2 x 2 diagonal matrices
is a subspace of M 2x2.
I know that a diagonal matrix is a square of n x n matrix whose
nondiagonal entries are zero, such as the n x n identity
matrix.
But could you explain every step of how to prove that this
diagonal matrix is a subspace of M 2x2.
Thanks.

Prove that if an m x m matrix A is upper-triangular, then
A-1 is also upper-triangular.
Hint: Obtain Axj=ej where xj is
the jth column of A-1 and ej is
the jth column of I. Then use back substitution to argue
that xij = 0 for i > j.

Prove that if A is a nonsingular nxn matrix, then so is cA for
every nonzero real number c.

How can I think about matrix entries in a general sense?
I a looking for a much deeper analysis than “systems of
equations.”
For instance, I know that when it is a rotation matrix, I know
that the column vectors in the matrix, R, will be where the basis
vectors land, and hence it will rotate any given vector
accordingly. (is this correct in a general sense, as far as
rotation matrices go?)
However, for a general linear transformation, I...

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...

The trace of a square n×nn×n matrix A=(aij)A=(aij) is the sum
a11+a22+⋯+anna11+a22+⋯+ann of the entries on its main
diagonal.
Let VV be the vector space of all 2×22×2 matrices with real
entries. Let HH be the set of all 2×22×2 matrices with real entries
that have trace 11. Is HH a subspace of the vector space
VV?
Does HH contain the zero vector of VV?
choose H contains the zero vector of V H does not contain the zero
vector...

Answer all of the questions true or false:
1.
a) If one row in an echelon form for an augmented matrix is [0 0 5
0 0]
b) A vector b is a linear combination of the columns of a matrix A
if and only if the
equation Ax=b has at least one solution.
c) The solution set of b is the set of all vectors of the form u =
+ p + vh
where vh is any solution...

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