Question

Let A=$\begin{pmatrix} 2 &-1 &-1 \\\ 1&0 &-1 \\\ 1&-1 &0 \end{pmatrix}$ and B=A–I. If ω=$\frac{\sqrt3 i-1}{2}$, then the number of elements in the set {n∈{1,2,⋯,100}:$A^n+(ωB)^n$=A+B} is equal to _______.

Answer 1

Here
A=$\begin{pmatrix} 2 &-1 &-1 \\\ 1&0 &-1 \\\ 1&-1 &0 \end{pmatrix}$
We get A 2 = A and similarly, for
B=A−I=$\begin{bmatrix} 1 &-1 &-1 \\\ 1& -1&-1 \\\ 1&-1 &-1 \end{bmatrix}$
We get,
B 2 = – B
⇒ B 3 = B
∴ A n + (ω B)n = A + (ω B)n for n ∈ N
For ω n to be unity n shall be multiple of 3 and for B n to be B.n shell be 3, 5, 7, … 99
∴ n = {3, 9, 15,….. 99}
Number of elements = 17.