Question

The sum of the elements in each row of \( A \in \mathbb{R}^{n \times n} \) is 1. If \( B = A^3 - 2A^2 + A \), which one of the following statements is correct (for \( x \in \mathbb{R}^n \))?

• A. The equation \( Bx = 0 \) has no solution
• B. The equation \( Bx = 0 \) has exactly two solutions
• C. The equation \( Bx = 0 \) has infinitely many solutions
• D. The equation \( Bx = 0 \) has a unique solution

Hint:

When analyzing matrices with a known eigenvector, check if the eigenvector is in the null space of the matrix after applying any polynomial transformations. This can help determine the number of solutions to linear systems involving the matrix.

Answer 1

The Correct Option is C

We are given that the sum of the elements in each row of \( A \) is 1, meaning the vector of all ones, \( \mathbf{1} \), is a right eigenvector of \( A \) corresponding to eigenvalue 1. In other words: \[ A\mathbf{1} = \mathbf{1}. \] Now, we consider the matrix \( B = A^3 - 2A^2 + A \). We can analyze the effect of applying \( B \) to \( \mathbf{1} \): \[ B\mathbf{1} = (A^3 - 2A^2 + A)\mathbf{1} = A^3\mathbf{1} - 2A^2\mathbf{1} + A\mathbf{1}. \] Since \( A\mathbf{1} = \mathbf{1} \), we find: \[ A^2\mathbf{1} = A\mathbf{1} = \mathbf{1}, \quad A^3\mathbf{1} = A\mathbf{1} = \mathbf{1}. \] Thus: \[ B\mathbf{1} = \mathbf{1} - 2\mathbf{1} + \mathbf{1} = 0. \] This shows that \( \mathbf{1} \) is in the null space of \( B \), meaning \( Bx = 0 \) has at least one non-trivial solution. Given that \( B \) is a matrix of degree 3, it is likely that the null space of \( B \) is of dimension greater than 1, implying that the equation \( Bx = 0 \) has infinitely many solutions.