Question

If A = `[(3/5, 2/5),(x, y)]` and A² = I, find x,y

Answer 1

Given
A = `[(3/5, 2/5),(x , y)]`

A² = A x A = `[(3/5, 2/5),(x , y)][(3/5, 2/5),(x , y)]`

= `[(9/25 + 2/5x, 6/25  2/5y),(3/5x + xy, 2/5x + y^2)]`

But A² = I = `[(1, 0),(0, 1)]`

`[(9/25 + 2/5x, 6/25  2/5y),(3/5x + xy, 2/5x + y^2)] = [(1, 0),(0, 1)]`
Comparing the corresponding elements,

`(9)/(25) + (2)/(5)x` = 1

⇒ `(2)/(5)x = 1 - (9)/(25) = (16)/(25)`

x = `(16)/(25) xx (5)/(2) = (8)/(5)`

`(6)/(25) + (2)/(5)y` = 0

⇒ `(2)/(5)y = (-6)/(25)`

y = `(-6)/(25) xx (5)/(2) = (-3)/(5)`

Hence x = `(8)/(5), y = (-3)/(5)`.