Question

MATRICES: How you determine x and y so that ... ?

`((0,x),(x,y))²` = `((x,x),(x,x+y))`

Answer 1

We have four possible solution

`x=0, y=0`
`x=0, y=1`
`x=1, y=0`
`x=1, y=1`

Explanation:

We seek `x` and `y` such that:

`( (0,x),(x,y) )^2 = ( (x,x),(x,x+y) )`

And so:

`( (0,x),(x,y) ) ( (0,x),(x,y) ) = ( (x,x),(x,x+y) )`

So then multiplying the two matrices together we have:

`( (0+x^2,0+xy),(0+xy,x^2+y^2) ) = ( (x,x),(x,x+y) )`

`:. ( (x^2,xy),(xy,x^2+y^2) ) = ( (x,x),(x,x+y) )`

If we equate elements we get four equations:

`[A]: x^2 = x`
`[B]: xy = x`
`[C]: xy = x`
`[D]: x^2+y^2 = x+y`

From Equation [A] we have:

`x^2=x => x^2-x= 0`
`:. x(x-1)= 0`
`:. x=0,1`

From Equation [B] (identical to [C]) we get:

`xy = x => xy - x =0`
`:. x(y-1) = 0`
`:. x=0,y=1`

From Equation [D] we get:

`x^2+y^2 = x+y`

When `x=0=> y^2=y => y(y-1) = 0 => y=0,1`
When `x=1=> 1+y^2=1+y => y^2=y => y=0,1`
When `y=1=> x^2+1=x+1 => x^2=x => x=0,1`

Thus, we have four possible solution

`x=0, y=0`
`x=0, y=1`
`x=1, y=0`
`x=1, y=1`