Question

If A = `[(2, 3),(1, 2)]` find x and y so that A² – xA + yI

If A = `[(2, 3),(1, 2)]` and A² = xA + yI, find the values of x and y.

Answer 1

Given
A² = `[(2, 3),(1, 2)][(2, 3),(1, 2)]`

= `[(2 xx 2 + 3 xx 1, 2 xx 3 + 3 xx 2),(1 xx 2 + 2 xx 1, 1 xx 3 + 2 xx 2)]`

= `[(4 + 3, 6 + 6),(2 + 2, 3 +4)]`

= `[(7, 12),(4, 7)]`
∵ A² = xA + yl

⇒ `[(7, 12),(4, 7)] = x[(2, 3),(1, 2)] + y[(1, 0),(0, 1)]`

⇒ `[(7, 12),(4, 7)] = [(2x, 3x),(x, 2x)] + [(y, 0),(0, y)]`

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

Comparing the corresponding elements

3x = 12

x = 4   ...(i)

2x + y = 7   ...(ii)

Substitute x = 4 into (1):

2x 4 + y = 7

8 + y = 7

y = 7 – 8

= –1

Hence x = 4, y = –1