Solve the following problems showing any necessary work This

Solve the following problems, showing any necessary work. This includes row operations. Find all ordered pairs (x, y) such that [x -1 2 y]^2 + [x y -2 1]^2 = -2 I_2 where I_2 is the 2 x 2 identity matrix, and A^2 = A. A.

Solution

( x 2 , -1 y)^2 = ( x^2-2 2x+2y , -x-y -2+y^2 )

(( x -2, 1 y )^2 = ( x^2 -2 -2x -2y , x +y -2 +y^2)

( x 2 , -1 y)^2 +(( x -2, 1 y )^2 = ( 2x^2 -4 0 , 0 -4+2y^2)

-2I = ( -2 0 , 0 -2 )

Now ( x 2 , -1 y)^2 +(( x -2, 1 y )^2 = ( -2 0 , 0 -2 )

( 2x^2 -4 0 , 0 -4+2y^2) = ( -2 0 , 0 -2 )

Equating the corresponding terms from LHS and RHS

we get \" 2x^2 -4 = -2 ----> 2x^2=2

x = + /- 1

Now -4 +2y^2 = -2

2y^2 =2

y = +/-1

Ordered Pairs ( x, y) ---> ( 1, 1) , ( -1, -1) , ( -1, 1) , and ( 1, -1)