Question

A matrix A ∈ Mn (R) is called nilpotent if there exists k∈N such that Aᵏ = On , the n×n zero matrix.

(a) Give an example of a 3×3 non-trivial nilpotent matrix A (i.e. A  = O₃) and the associated smallest k such that Aᵏ = O₃.

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Answer 1

An example of a 3×3 non-trivial nilpotent matrix is A = [[0, 1, 0], [0, 0, 1], [0, 0, 0]], with the smallest k being 3 since A³ results in the zero matrix.

Step-by-step explanation:

A nilpotent matrix is a matrix that can be raised to a power resulting in the zero matrix. For a 3×3 example, consider the following non-trivial nilpotent matrix A:

A = [0 1 0]
[0 0 1]
[0 0 0]

Now, let's find the smallest k such that Aᵗ = O₃. If we calculate A², we get:

A² = [0 0 1]
[0 0 0]
[0 0 0]

And if we further calculate A³, we will have:

A³ = [0 0 0]
[0 0 0]
[0 0 0]

Thus, A³ = O₃, and the smallest k for which Aᵗ = O₃ is k = 3.