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Matrix algebra

1.
Given the matrices

[010] [100] [100]
A=[101] B= 010] C=[000]
[010] [001] [00-1]

Show that A and B commute, B and C commute but A and C do not.

2.
Show that the matrix

[1 4 0]
[2 5 0]
[3 6 0]

Is not invertible

3.
Find the inverse of the matrix

[1 2 3]
[2 5 3]
[1 0 8]

And confirm your result by direct calculation.

4.
Let A,B be aritrary matrices whose product exists.

Show that AA^ and A^A exist and Hermitian (^ represent complex conjugate)
Show that the product (B^A^) exists and equal to (AB)^
If both A and B are Hermitian, show that AB+BA is Hermitian as well
If A and B are both Hermitian, show that (i(AB-BA) is Hermitian as well.

5.
show that each of the Pauli spin matrices is both Hermitian and unitary. Calculate teh inverse of each.
Show that the product of two Pauli matrices is (see file for complete formulation)

6.
Let A, B and C be square matrices.
show that

tr(AB) = tr(BA)
tr(ABC)=tr(BCA)=tr(CAB)

is tr(ACB)=tr(ABC)?

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