Question

Consider a square matrix ( A ) in which the sum of the entries of each row equals the same scalar ( r ). To show that ( r ) is an eigenvalue of ( A ), which of the following steps can be taken?

- A) Prove that ( r ) is an eigenvalue by finding an eigenvector.
- B) Prove that ( r ) is an eigenvalue by examining the determinant of ( A ).
- C) Prove that ( r ) is an eigenvalue by checking the trace of ( A ).
- D) Prove that ( r ) is an eigenvalue by solving the characteristic equation of ( A ).

Answer 1

To prove that (r) is an eigenvalue of a square matrix (A), finding an eigenvector corresponding to (r) or solving the characteristic equation of (A) is effective. Checking the trace or determinant alone does not confirm the existence of a specific eigenvalue.

Step-by-step explanation:

To show that a scalar (r) is an eigenvalue of a square matrix (A) where the sum of the entries of each row is equal to (r), one of the effective methods would be to find an eigenvector corresponding to (r). This can be done by considering a vector with each entry being 1, which will be an eigenvector if (r) is indeed an eigenvalue. This is because multiplying A by this vector will result in a vector where each entry is the sum of the rows of A, which we know equals (r).

However, checking the trace or the determinant of A will not directly show that (r) is an eigenvalue. Specifically, the determinant is related to the product of the eigenvalues, and the trace is the sum of the eigenvalues, but neither guarantees the existence of a specific eigenvalue. Lastly, solving the characteristic equation of A would certainly confirm whether (r) is an eigenvalue or not, as the roots of the characteristic equation are the eigenvalues of A.