To analyze this problem, we will address the key components step by step based on the provided matrix $A$ and the associated details:

Part a) Eigenvalues and Associated Properties

1. Distinct Eigenvalues:
   From the text, it appears that the matrix $A$ has only one distinct eigenvalue, 0, with multiplicity 1 and eigenspace dimension 1.

   However, let's verify this by calculating the eigenvalues:

   • The eigenvalues are roots of the characteristic polynomial $\text{det}(A - \lambda I) = 0$.
   • Solving this will confirm whether the eigenvalue $0$ is correct and check its algebraic and geometric multiplicities.

2. Multiplicity and Dimensions:

   • Algebraic Multiplicity: The number of times $0$ appears as a root of the characteristic polynomial. Here, it's given as 1.
   • Geometric Multiplicity: The dimension of the eigenspace associated with $0$, which is also stated as 1.

Part b) Diagonalizability of $A$

A matrix is diagonalizable if the algebraic multiplicity of each eigenvalue matches its geometric multiplicity and the sum of geometric multiplicities equals the matrix's size.

• Here, $A$ has one eigenvalue, $0$, with:
  • Algebraic Multiplicity = 1.
  • Geometric Multiplicity = 1.
• Since these match, $A$ is diagonalizable.

Conclusion:

The matrix $A$ is diagonalizable, as the geometric multiplicities match the algebraic multiplicities for all eigenvalues.

Would you like me to perform explicit calculations for the eigenvalues, eigenspaces, or verify diagonalizability in detail?

Expanding Questions:

1. What are the steps to compute the characteristic polynomial of a matrix?
2. How do algebraic and geometric multiplicities affect diagonalizability?
3. Why is the eigenspace dimension important in determining diagonalizability?
4. Could a matrix with only one distinct eigenvalue ever fail to be diagonalizable?
5. What are the practical implications of a matrix being diagonalizable?

Tip: To confirm diagonalizability, always compare the sum of eigenspace dimensions to the matrix's size.