The question is asking to find the rank of a $5 \times 5$ diagonalizable matrix $A$ given its eigenvalues: $0, 0, 0, 5, 2$. Here's how to solve it:

Step 1: Understand the Eigenvalues and Rank

The eigenvalues of a matrix determine its rank because:

• The rank of a matrix is the number of non-zero rows in its row-reduced echelon form.
• The number of non-zero eigenvalues of a diagonalizable matrix corresponds to the rank of the matrix.

Step 2: Count Non-Zero Eigenvalues

From the given eigenvalues $0, 0, 0, 5, 2$:

• There are two non-zero eigenvalues ($5$ and $2$).

Step 3: Rank of $A$

Since there are 2 non-zero eigenvalues, the rank of $A$ is 2.

Final Answer:

$\text{rank}(A) = 2$

Would you like me to explain this further or discuss related concepts? Here are some follow-up questions you might find useful:

1. Why does the number of non-zero eigenvalues correspond to the rank of the matrix?
2. How does diagonalizability of a matrix affect the interpretation of eigenvalues?
3. What happens to the rank of $A$ if one more eigenvalue is $0$?
4. Can a non-diagonalizable matrix have the same rank in this case? Why or why not?
5. What is the connection between the nullity and rank of a matrix in this context?

Tip: Always check the eigenvalues and multiplicities when dealing with rank and nullity problems in diagonalizable matrices!