Matrices · Question 17

Let S = \m Z : Am2 + Am = 3I - A-6 \, where A = bmatrix 2 & -1 \ 1 & 0 bmatrix. Then n(S) is equal to .

Answer

2

Solution

Let N = A - I = bmatrix 1 & -1 \ 1 & -1 bmatrix

N2 = bmatrix 1 - 1 & -1 + 1 \ 1 - 1 & -1 + 1 bmatrix = O

Ak = (I + N)k = I + kN for k N (binomial expansion, N2 = O)

Also (I + N)(I - N) = I - N2 = I A-1 = I - N, so Ak = I + kN for every integer k

Am2 + Am = 2I + (m2 + m)N

3I - A-6 = 3I - (I - 6N) = 2I + 6N

N O m2 + m = 6 (m + 3)(m - 2) = 0

m = 2 or m = -3

n(S) = 2

Answer: 2