MATRICES: For which value of x and y ... ?

Question

For which value of x and y is the product of the matrix #((1,x),(0,1))# with the matrix #((1,0),(y,1))# commutative?

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Answer 1 · 2017-10-10T16:49:36

See below.

Calling #A = ((1,x),(0,1))# and #B=((1,0),(y,1))#

they commute if

#A cdot B = B cdotA#

or

#A cdot B-B cdot A = ((x y, 0),(0, -x y))=0_2#

Then they commute if #x = 0# or #y = 0# or #x=y=0#

NOTE:

#A cdot B = ((1 + x y, x),(y, 1))#
#B cdot A = ((1, x),(y, 1 + x y))#

then

#A cdot B-B cdot A = ((x y, 0),(0, -x y))#