Question

Find the 4 by 4 cyclic permutation matrix: (x1, x2, x3, x4) is transformed to ax = (x2, x3, x4, x1). What is the effect of a²? Show that a³ = a^(-1).

a. a² transforms (x1, x2, x3, x4) to (x3, x4, x1, x2), a³ = a^(-1)
b. a² transforms (x1, x2, x3, x4) to (x4, x1, x2, x3), a³ = a^(-1)
c. a² transforms (x1, x2, x3, x4) to (x1, x2, x3, x4), a³ = a^(-1)
d. a² transforms (x1, x2, x3, x4) to (x2, x3, x4, x1), a³ = a^(-1)

Answer 1

The 4 by 4 cyclic permutation matrix is found by shifting each element one position to the right. The effect of a² is obtained by multiplying the matrix a by itself. The matrix a³ is equal to the inverse of a.

Step-by-step explanation:

To find the 4 by 4 cyclic permutation matrix, we can observe that each element in the original matrix is shifted one position to the right. Starting with (x1, x2, x3, x4), the transformation ax = (x2, x3, x4, x1) can be represented as:

a = [[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [1, 0, 0, 0]]

To find the effect of a², we can multiply the matrix a by itself:

a² = a ⋅ a = [[0, 0, 1, 0], [0, 0, 0, 1], [1, 0, 0, 0], [0, 1, 0, 0]]

Now let's calculate a³:

a³ = a ⋅ a² = [[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [1, 0, 0, 0]] ⋅ [[0, 0, 1, 0], [0, 0, 0, 1], [1, 0, 0, 0], [0, 1, 0, 0]] = [[0, 0, 0, 1], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]]

Finally, we can show that a³ is equal to the inverse of a:

a³ = a^(-1)

Therefore, the correct answer is option d. a² transforms (x1, x2, x3, x4) to (x2, x3, x4, x1), a³ = a^(-1).