To solve the problem described in the text regarding nilpotent matrices, let’s address the two points mentioned:

Definitions and General Approach

A matrix $A \in \mathbb{F}^{n \times n}$ is nilpotent if $A^k = 0$ for some $k \geq 1$, where $k$ is the smallest integer for which $A^k = 0$.

The provided matrix $A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$ is given as an example. For this matrix:

1. $A^2 \neq 0$,
2. $A^3 = 0$.

We are asked to address two proofs about nilpotent matrices.

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1. Proof that Every Nilpotent Matrix is Singular

Claim: Every nilpotent matrix $A$ is singular (i.e., $\det(A) = 0$).

• If $A$ is nilpotent, there exists $k \geq 1$ such that $A^k = 0$.
• The determinant of a matrix satisfies $\det(A^k) = (\det(A))^k$.
• Since $A^k = 0$, we have $\det(A^k) = 0$.
• Therefore, $(\det(A))^k = 0$, which implies $\det(A) = 0$.

Thus, $A$ is singular.

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2. Proof That a Nilpotent Matrix is Not Invertible

Claim: A nilpotent matrix $A$ cannot be invertible.

• Assume $A$ is nilpotent, so $A^k = 0$ for some $k \geq 1$.
• If $A$ were invertible, there would exist an inverse $A^{-1}$ such that $A \cdot A^{-1} = I$, where $I$ is the identity matrix.
• Multiplying both sides of $A^k = 0$ by $A^{-1}$, we would get: $A^{k-1} \cdot A \cdot A^{-1} = A^{k-1} \cdot I = 0,$ which contradicts the assumption $A^k = 0$ unless $A = 0$ (which is trivial).
• Thus, $A$ is not invertible.

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Summary of Results

1. Every nilpotent matrix is singular ($\det(A) = 0$).
2. Every nilpotent matrix is not invertible.

Let me know if you would like further clarification or have additional questions.

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Relative Questions

1. Can you compute the eigenvalues of a nilpotent matrix? What are their properties?
2. What is the geometric interpretation of nilpotent matrices in terms of linear transformations?
3. How do the powers of nilpotent matrices relate to their Jordan canonical form?
4. Can you provide examples of nilpotent matrices with higher orders (e.g., $k > 3$)?
5. How does nilpotency affect the trace of a matrix?

Tip

Nilpotent matrices always have all eigenvalues equal to 0, which can be a helpful property when analyzing