Question

[A] = [x] [B] [x]

[x] = ???

[A], [B], and [x] are 2x2 matrices, and [A] and [B] are known. find [x] in terms of [A] and [B]

they are invertible. state any other assumltions

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.

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

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

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?

Suppose A and B are invertible matrices in Mn(R) and
that A + B is also invertible. Show that C = A-1 +
B-1 is also invertible.

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 G be the set of all 2x2 matrices [a a a a] such that a is in
the reals and a does not equal 0.
Prove or disprove that G is a group under matrix
multiplication.

A and B are two m*n matrices. a. Show that B is invertible. b.
Show that Nullsp(A)=Nullsp(BA)

1. Find the orthogonal projection of the matrix
[[3,2][4,5]] onto the space of diagonal 2x2 matrices of the form
lambda?I.
[[4.5,0][0,4.5]] [[5.5,0][0,5.5]] [[4,0][0,4]] [[3.5,0][0,3.5]] [[5,0][0,5]] [[1.5,0][0,1.5]]
2. Find the orthogonal projection of the matrix
[[2,1][2,6]] onto the space of symmetric 2x2 matrices of trace
0.
[[-1,3][3,1]] [[1.5,1][1,-1.5]] [[0,4][4,0]] [[3,3.5][3.5,-3]] [[0,1.5][1.5,0]] [[-2,1.5][1.5,2]] [[0.5,4.5][4.5,-0.5]] [[-1,6][6,1]] [[0,3.5][3.5,0]] [[-1.5,3.5][3.5,1.5]]
3. Find the orthogonal projection of the matrix
[[1,5][1,2]] onto the space of anti-symmetric 2x2
matrices.
[[0,-1] [1,0]] [[0,2] [-2,0]] [[0,-1.5]
[1.5,0]] [[0,2.5] [-2.5,0]] [[0,0]
[0,0]] [[0,-0.5] [0.5,0]] [[0,1] [-1,0]]
[[0,1.5] [-1.5,0]] [[0,-2.5]
[2.5,0]] [[0,0.5] [-0.5,0]]
4. Let p be the orthogonal projection of
u=[40,-9,91]T onto the...

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