Question

We have matrix `A=[(2,x),(y,3)]` and `B=[(y,-3),(5,2x)]`, `A,BinMM_(2,2) (CC)`. How you find `x,y` such that `(A+xI_2)(B+yI_2)=[(-25,59),(2x+y^2,20)]` ?

Answer 1

Given: `A=[(2,x),(y,3)]` and `B=[(y,-3),(5,2x)]`

Compute `A+xI_2`:

`A+xI_2 = [(2,x),(y,3)] + x[(1,0),(0,1)]`

`A+xI_2 = [(2,x),(y,3)] + [(x,0),(0,x)]`

`A+xI_2 = [(2+x,x),(y,3+x)]" [1]"`

Compute `B+yI_2`

`B+yI_2 = [(y,-3),(5,2x)] + y[(1,0),(0,1)]`

`B+yI_2 = [(y,-3),(5,2x)] + [(y,0),(0,y)]`

`B+yI_2 = [(2y,-3),(5,2x+y)]" [2]"`

Substitute equation [1] and equation [2] into the given equation `(A+xI_2)(B+yI_2)=[(-25,59),(2x+y^2,20)]`:

`[(2+x,x),(y,3+x)][(2y,-3),(5,2x+y)]=[(-25,59),(2x+y^2,20)]`

Performing the multiplication:

`[(5x+2xy+2y,2x^2 +xy-3x-6),(2y^2+5x+15,2x^2+xy+6x)]=[(-25,59),(2x+y^2,20)]`

We have two unknown values and we have 4 equation from which we may chose; let's use the two equations in column 2:

`2x^2 +xy-3x-6 = 59" [3]"`
`2x^2+xy+6x = 20" [4]"`

Subtract equation [4] from equation [3]:

`-9x-6 = 39`

`-9x = 45`

`x = -5`

Substitute -5 for x into equation [4]:

`2(-5)^2-5y+6(-5) = 20`

`-5y = 0`

`y = 0`