Question

If \( P \) is a non-singular matrix of order \( 5 \times 5 \) 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. \( \frac{1}{5} \)
• D. \( 5 \)

Hint:

Consider the action of the matrix on a vector of ones to represent row sums.

Answer 1

The Correct Option is B

Step 1: Represent the row sum condition as a vector.
\nDefine \( \mathbf{1} = \begin{pmatrix} 1
1
1
1
1 \end{pmatrix} \). The condition becomes \( P\mathbf{1} = \mathbf{1} \).\n\n
Step 2: Multiply by \( P^{-1} \).
\n\( P^{-1}(P\mathbf{1}) = P^{-1}\mathbf{1} \Rightarrow I\mathbf{1} = P^{-1}\mathbf{1} \Rightarrow \mathbf{1} = P^{-1}\mathbf{1} \).\n\n
Step 3: Understand \( P^{-1}\mathbf{1} = \mathbf{1} \).
\nIf \( P^{-1} = (a_{ij}) \), the \( i \)-th component of \( P^{-1}\mathbf{1} \) is \( \sum_{j=1}^{5} a_{ij} \).
\nThe equation \( P^{-1}\mathbf{1} = \mathbf{1} \) means the sum of the elements in each row of \( P^{-1} \) is 1.