Let A = bmatrix 2 & -2 & -4 -1 & 3 & 4 1 & -2 & -3 bmatrix and B = I - A , where I is the identity matrix of order 3 . Let z₁ = 1+i 2 and z₂ = -1+ 3 i 2 . If M = z₁ A + z₂ B , then the number of integers n 1, 2, , 100 such that M^n = I is e

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Let A = bmatrix 2 & -2 & -4 -1 & 3 & 4 1 & -2 & -3 bmatrix and B = I - A , where I is the identity matrix of order 3 . Let z₁ = 1+i 2 and z₂ = -1+ 3 i 2 . If M = z₁ A + z₂ B , then the number of integers n 1, 2, , 100 such that M^n = I is equal to _____ .

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Correct answer: 4

First, we compute A^2 : A^2 = bmatrix 2 & -2 & -4 -1 & 3 & 4 1 & -2 & -3 bmatrix bmatrix 2 & -2 & -4 -1 & 3 & 4 1 & -2 & -3 bmatrix = bmatrix 2 & -2 & -4 -1 & 3 & 4 1 & -2 & -3 bmatrix = A Since A^2 = A , A is idempotent. For B = I - A , we have: B^2 = (I - A)^2 = I - 2A + A^2 = I - A = B Also, AB = A(I - A) = A - A^2 = 0 and BA = (I - A)A = A - A^2 = 0 . Given M = z₁ A + z₂ B . Since AB = BA = 0 , the cross terms in the expansion of M^n vanish, and we get: M^n = z₁^n A^n + z₂^n B^n Because A and B are idempotent,

Let A = bmatrix 2 & -2 & -4 -1 & 3 & 4 1 & -2 & -3 bmatrix and B = I - A , where I is the identity matrix of order 3 . Let z₁ = 1+i 2 and z₂ = -1+ 3 i 2 . If M = z₁ A + z₂ B , then the number of integers n 1, 2, , 100 such that M^n = I is equal to _____ .

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