Question

Question #0ed08

Answer 1

`(x, y) = (-4, 1)`

Explanation:

Performing the multiplication, first, we have

`A^2=((3,2),(1,1))^2 = ((3,2),(1,1))((3,2),(1,1))=((11,8),(4,3))`

`xA = x((3,2),(1,1)) = ((3x,2x),(x,x))`

`yI = y((1,0),(0,1)) = ((y,0),(0,y))`

Then, summing, we have

`A^2+xA+yI = ((11,8),(4,3))+((3x,2x),(x,x))+((y,0),(0,y))`

`=((3x+y+11,2x+8),(x+4,x+y+3)) = ((0,0),(0,0))`

Equating entries with corresponding indices, we get the following system of equations from the bottom row:

`{(x+4 = 0), (x+y+3 = 0):}`

From the first equation, we have `x = -4`

Substituting this into the second equation, we get

`-4+y+3=0`

`=> y = 1`

If we substitute these values in for `x` and `y` we find that they fulfill the given equation, and thus we have `(x, y) = (-4, 1)`