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

#((0,x),(x,y))²# = #((x,x),(x,x+y))#

1 Answer
Oct 10, 2017

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 #