Can Non-Commutative Symmetric Matrices Defy Mathematical Rules?

[brandy]

hey i was just thinking how matrices aren't supposed to be commutative?
well i was just thinking, what if the matricees were equal? and i was like ha! I am so cool! take that mr logic.
so yea, so far i have a couple rebelious matricees
a*b where a=1 by 1 and b= 1 by 1
a*b where na=b
a*b where a^n=b
a*b where a= identity
a*b where na=b and a = identity
a*b where a = null
a*b where a= mutliple of identity and b= multiple of mirrored identity

so, my questiono is this, can you help me be a good little poindexter and come up with a couple others?

 

[VeeEight]

Consider what happens if B = A + xI (for some real number x, I being the identity matrix) and if A and B are diagonal matrices.

 

[mathman]

Both A and B symmetric.