Question

Let A be a 5×5 matrix such that the sum of the entries of each row is equal to 5. Select a number that μst be an eigenvalue of A.

a) 1
b) 5
c) 0
d) 25

Answer 1

5 must be an eigenvalue of matrix A because if you multiply A by a vector with all entries of 1, the resulting vector will have entries equal to the row sum of matrix A, which is known to be 5. Therefore, Av = 5v, indicating 5 is an eigenvalue.

Step-by-step explanation:

The question asks which number must be an eigenvalue of matrix A, knowing that A is a 5×5 matrix and the sum of the entries of each row is equal to 5. If we take a vector v whose all entries are 1, when we multiply A by this vector v, we will get a new vector where each entry is the sum of the entries of each corresponding row of A, which is known to be 5. This means Av = 5v, which shows that 5 is an eigenvalue of A since v is nonzero. Hence, the correct answer is b) 5.