Question

If A = `[[3,1] , [7,5]]`, find the values of x and y such that A² + xI₂ = yA.

Answer 1

Given that: A = `[[3,1] , [7,5]]`

Now,  A²  = A.A = `[[3,1] , [7,5]] [[3,1] , [7,5]]`

= `[[9 +7, 3+5], [ 21 + 35, 7 + 25]]`

= `[[16,8], [56,32]]`  

∴ A² + xI₂ = yA 

`[[16,8], [56,32]] + x [[ 1,0], [0,1]] = y [[3,1] ,[7,5]]`

`[[16 +x ,8],[ 56 , 32+x]] = [[3y, y],[7y,5y]]`

Comparing the corresponding entries of equal matrices, we have ⇒ y = 8

and 16 + x = 3y
∴  x = 3 x 8 - 16
       = 24 - 16
       = 8

Hence, the required values of x is 8 and y is 8.