Gate DA 2025 | Question - 28 | Linear Algebra

🔍 Step 1: Understanding $A = I_n + xx^\top$

• $xx^\top$ is a rank-1 symmetric matrix.

• Since $x^\top x = 1$, it’s a unit vector, so $xx^\top$ is a positive semi-definite matrix.

• $I_n$ is identity matrix ⇒ positive definite.

So,

• $A = I_n + xx^\top$ is positive definite.

• All eigenvalues of $A$ are positive.

Option-wise Analysis:

(A) Rank of $A$ is $n$✅ Correct

• $\text{Rank}(I_n) = n$

• $\text{Rank}(xx^\top) = 1$, but adding it to $I_n$ does not reduce rank.

• So, $\text{Rank}(A) = n$

(B) $A$ is invertible ✅ Correct

• Since all eigenvalues of $A$ are positive, it’s positive definite ⇒ invertible.

(C) 0 is an eigenvalue of $A$❌ Incorrect

• A positive definite matrix cannot have a zero eigenvalue. All eigenvalues are > 0.

(D) $A^{-1}$ has a negative eigenvalue ❌ Incorrect

• The inverse of a positive definite matrix is also positive definite.

• So all eigenvalues of $A^{-1}$ are positive, not negative.