Question

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

Answer #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 K be a positive definite matrix. Prove that K is invertible,
and that K^(-1) is also positive definite.

Assume A is an invertible matrix
1. prove that 0 is not an eigenvalue of A
2. assume λ is an eigenvalue of A. Show that λ^(-1) is an
eigenvalue of A^(-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

Give an example of a nondiagonal 2x2 matrix that is
diagonalizable but not invertible. Show that these two facts are
the case for your example.

a. Let T : V → W be left invertible. Show that T is
injective.
b. Let T : V → W be right invertible. Show that T is
surjective

Show/Prove that every invertible square (2x2) matrix is a
product of at most four elementary matrices

A square matrix A is said to be idempotent if A2 = A. Let A be
an idempotent matrix.
Show that I − A is also idempotent.
Show that if A is invertible, then A = I.
Show that the only possible eigenvalues of A are 0 and 1.(Hint:
Suppose x is an eigenvector with associated eigenvalue λ and then
multiply x on the left by A twice.) Let W = col(A).
Show that TA(x) = projW x and TI−A(x)...

A square matrix A is said to be idempotent if
A2 = A. Let A be an
idempotent matrix.
Show that I − A is also
idempotent.
Show that if A is invertible, then A =
I.
Show that the only possible eigenvalues of A are 0 and
1.(Hint: Suppose x is an eigenvector with
associated eigenvalue λ and then multiply
x on the left by A twice.)
Let W = col(A). Show that
TA(x) =
projW x and
TI−A(x)...

1.4.3. Let T(x) =Ax+b be an invertible affine transformation of
R3.
Show that T ^-1 is also affine.

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