Question

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

Answer #1

Prove or disprove: GL2(R), the set of invertible 2x2 matrices,
with operations of matrix addition and matrix multiplication is a
ring.
Prove or disprove: (Z5,+, .), the set of invertible
2x2 matrices, with operations of matrix addition and matrix
multiplication is a ring.

prove that type 1 elementary matrix is a product of type 2 and 3
elementary matrices

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.

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 any Givens rotator matrix in R2 is a product of two
Householder reflector matrices. Can a Householder reflector matrix
be a product of Givens rotator matrices?

(7) Prove the following statements.
(c) If A is invertible and similar to B, then B is invertible
and A−1 is similar to B−1 .
(d) The trace of a square matrix is the sum of the diagonal
entries in A and is denoted by tr A. It can be verified that tr(F
G)=tr(GF) for any two n × n matrices F and G. Prove that if A and B
are similar, then tr A = tr B

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)

a).For the reduction of matrix determine the elementary matrices
corresponding to each operation. M= 1 0 2 1 5
1 1 5 2 7
1 2 8 4 12 b). Calculate the product P of these elementary
matrices and verify that PM is the end result.

If I prove Det(A)Det(B) = Det(AB) for matrices A and B when A
and B are 2x2 matrices, can I use that to show that Det(A)Det(B) =
Det(AB) for any n x n matrix? If so how?

Show that if a square matrix K over Zp ( p prime) is
involutory ( or self-inverse), then det K=+-1
(An nxn matrix K is called involutory if K is invertible and
K-1 = K)
from Applied algebra
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