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

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