The problem asks to prove that if $A$ is a symmetric and idempotent matrix of rank $r$, then the trace of $A$, denoted by $\text{tr}(A)$, equals the rank $r$, i.e., $\text{tr}(A) = r$.

Proof Outline:

Given:

• $A$ is symmetric: $A = A^T$.
• $A$ is idempotent: $A^2 = A$.
• The rank of $A$ is $r$.

To Prove:

$\text{tr}(A) = r.$

Step-by-Step Proof:

1. Idempotency: The condition $A^2 = A$ implies that all eigenvalues of $A$ must be either 0 or 1.

   • Explanation: Let $\lambda$ be an eigenvalue of $A$. From $A^2 = A$, we get $\lambda^2 = \lambda$, which leads to $\lambda = 0$ or $\lambda = 1$.

2. Diagonalizability: Since $A$ is symmetric, it is diagonalizable. Thus, $A$ can be written as: $A = P \Lambda P^{-1},$ where $\Lambda$ is a diagonal matrix containing the eigenvalues of $A$, and $P$ is an orthogonal matrix containing the eigenvectors.

3. Rank and Eigenvalues: The rank of a matrix is the number of non-zero eigenvalues. Since the eigenvalues of $A$ are 0 or 1, the rank of $A$, denoted by $r$, is the number of 1's in the diagonalized form of $A$.

4. Trace of a Matrix: The trace of a matrix is the sum of its eigenvalues. Since the eigenvalues of $A$ are 0 or 1, and the rank $r$ corresponds to the number of eigenvalues equal to 1, we conclude that: $\text{tr}(A) = \sum_{i=1}^{n} \lambda_i = r,$ where $\lambda_i$ are the eigenvalues of $A$ and $r$ is the number of eigenvalues equal to 1.

Thus, we have proven that the trace of $A$ is equal to its rank: $\text{tr}(A) = r.$

Let me know if you'd like further details on any part of this proof.

Here are five related questions to deepen your understanding:

1. What properties do idempotent matrices possess beyond having eigenvalues of 0 or 1?
2. Can the trace of any symmetric matrix ever exceed its rank?
3. How does the concept of diagonalization simplify proofs involving symmetric matrices?
4. How is the rank of a matrix related to its nullity in the context of the Rank-Nullity Theorem?
5. Can an idempotent matrix be non-symmetric and still satisfy $A^2 = A$?

Tip: When dealing with symmetric matrices, always check if the matrix can be diagonalized easily—this can significantly simplify the computation of eigenvalues and other matrix properties.