Question

If A = `[(3, 2),(1, 1)]` and A² + Ax + yI = 0, find the values of x and y.

Answer 1

A² + Ax + yI = 0

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

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

= `[(9 + 2, 6 + 2),(3 + 1, 2 + 1)]`

= `[(11, 8),(4, 3)]`

A² + Ax + yI = 0

⇒ `[(11, 8),(4, 3)] + x xx [(3, 2),(1, 1)] + y xx [(1, 0),(0, 1)] = [(0, 0),(0, 0)]`

⇒ `[(11, 8),(4, 3)] + [(3x, 2x),(x, x)] + [(y, 0),(0, y)] = [(0, 0),(0, 0)]`

⇒ `[(11 + 3x + y, 8 + 2x),(4 + x, 3 + x + y)] = [(0, 0),(0, 0)]`

Comparing the corresponding elements

8 + 2x = 0

2x = −8

x = `(−8)/2`

x = −4

11 + 3x + y = 0

Substitute x = −4

11 + 3(−4) + y = 0

11 − 12 + y = 0

y = 1

Hence  x = −4, y = 1