Question

Let \( P \) be a \( 7 \times 7 \) matrix of rank 4 with real entries. Let \( a \in \mathbb{R}^7 \) be a column vector. Then the rank of \( P + aa^T \) is at least ..........

Hint:

For matrices of rank \( r \), adding a rank-1 matrix (like \( aa^T \)) can increase the rank by at most 1, but it cannot decrease the rank.

Answer 1

Correct Answer: 2.9 - 3.1

Step 1: Understanding the rank of \( P + aa^T \).
We are given that \( P \) is a \( 7 \times 7 \) matrix of rank 4, and \( a \) is a column vector in \( \mathbb{R}^7 \). The rank of the matrix \( P + aa^T \) depends on the rank of \( P \) and the rank of the outer product \( aa^T \).
Step 2: Rank of \( aa^T \).
The rank of the matrix \( aa^T \) is 1, as it is the outer product of a vector with itself. The matrix \( aa^T \) has at most rank 1, regardless of the dimension of \( a \).
Step 3: Rank of the sum.
Since \( P \) has rank 4, and the rank of \( aa^T \) is 1, the rank of the sum \( P + aa^T \) is at least 4. This is because adding a rank 1 matrix to a matrix of rank 4 does not decrease the rank unless the vector \( a \) is in the null space of \( P \), which we do not assume here.
Step 4: Conclusion.
Thus, the rank of \( P + aa^T \) is at least 4.