Question

For which values of constants a, b, and c are the matrices in exercises 40 through 50 diagonalizable?

a) a = 0, b = 3, c = 1
b) a = 1, b = 0, c = 3
c) a = 3, b = 1, c = 0
d) a = 0, b = 1, c = 3

Answer 1

To determine the values of constants a, b, and c for which the matrices in exercises 40 through 50 are diagonalizable, we can use the characteristic equation to find the eigenvalues and check if they are distinct.

Step-by-step explanation:

To determine the values of constants a, b, and c for which the matrices in exercises 40 through 50 are diagonalizable, we can use the following steps:

1. For a matrix to be diagonalizable, it must have the same number of distinct eigenvalues as its dimension.
2. We can find the eigenvalues of a matrix by solving the characteristic equation. In this case, the characteristic equation is defined as:
3. det(A - λI) = 0
4. Substitute the values of a, b, and c into the matrix A and solve the characteristic equation to find the eigenvalues.
5. If the number of distinct eigenvalues is equal to the dimension of the matrix, then the matrix is diagonalizable for those values of a, b, and c.