Question

For the matrix A = `[(3,2),(1,1)]` find the numbers a and b such that A² + aA + bI = 0.

Answer 1

We are given that A = `[(3,2),(1,1)]`

Now A² + aA + bI = 0

`[(3,2),(1,1)] [(3,2),(1,1)] + a[(3,2),(1,1)] + b[(1,0),(0,1)] = [(0,0),(0,0)]`

⇒ `[(9 + 2, 6 + 2),(3 + 1,2 + 1)] + [(3a, 2a),(a,a)] + [(b,0),(0,b)] = [(0,0),(0,0)]`

⇒ `[(11 + 3a + b, 8 + 2a + 0),(4 + a + 0, 3 + a + b)] = [(0,0),(0,0)]`

⇒ 4 + a = 0

⇒ a = −4

Also, 3 + a + b = 0

⇒ 3 − 4 + b = 0

⇒ b = 1

∴ a = −4, b = 1