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