Question

If \( P \) is a non-singular matrix of order \( 5\times5 \) and the sum of the elements of each row is 1, then the sum of the elements of each row in \( P^{-1} \) is:

• A. \( 0 \)
• B. \( 1 \)
• C. \( \frac{1}{8} \)
• D. \( 8 \)

Hint:

Row sums: Vector of ones is eigenvector. Inverse preserves eigenvector.

Answer 1

The Correct Option is B

Concept: Row sum = 1 ⇒ vector of ones is eigenvector. Step 1: Let \[ e = (1,1,\dots,1)^T \] Then: \[ Pe = e \] Step 2: Multiply by the inverse: \[ P^{-1} Pe = P^{-1} e \Rightarrow e = P^{-1} e \] Thus, the row sums of \( P^{-1} \) are also 1.