Question

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

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?

Answer 1

See below.

Explanation:

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))`