To find the values of a and b, we need to use the properties of eigenvalues and eigenvectors.

Properties of Eigenvalues:
1. The sum of the eigenvalues of a matrix is equal to the trace of the matrix.
2. The product of the eigenvalues of a matrix is equal to the determinant of the matrix.
3. If λ is an eigenvalue of a matrix A, then λ^k is an eigenvalue of A^k, where k is a positive integer.

Properties of Eigenvectors:
1. If λ is an eigenvalue of a matrix A, then the eigenvector corresponding to λ is a non-zero vector in the null space of (A - λI), where I is the identity matrix.
2. Eigenvectors corresponding to different eigenvalues are linearly independent.

Now let's solve the problem step by step:

Step 1: Find the remaining eigenvalues of M.
Since we are given that 2 and -1 are eigenvalues of M, we can use the properties of eigenvalues to find the sum and product of all eigenvalues.

Sum of eigenvalues = trace(M)
Product of eigenvalues = det(M)

From the properties, we can write:
2 + (-1) + λ3 + λ4 + λ5 + λ6 = trace(M)
2 * (-1) * λ3 * λ4 * λ5 * λ6 = det(M)

Step 2: Determine the values of λ3, λ4, λ5, and λ6.
From the given information, we know that M is a 6x6 real matrix. Therefore, the sum of eigenvalues is equal to the trace of M, which is the sum of the diagonal elements of M.

Let's assume the diagonal elements of M are a1, a2, a3, a4, a5, and a6.

2 + (-1) + λ3 + λ4 + λ5 + λ6 = a1 + a2 + a3 + a4 + a5 + a6

Since the trace of M is equal to the sum of eigenvalues, we can substitute the diagonal elements of M into the equation:

2 + (-1) + λ3 + λ4 + λ5 + λ6 = a1 + a2 + a3 + a4 + a5 + a6

From this equation, we can see that the remaining eigenvalues are a1, a2, a3, a4, a5, and a6.

Step 3: Find the value of a.
To find the value of a, we need to calculate M^5.

Using the properties of eigenvectors, we know that M^5 can be written as:

M^5 = (a1^5) * v1 + (a2^5) * v2 + (a3^5) * v3 + (a4^5) * v4 + (a5^5) * v5 + (a6^5) * v6

where v1, v2, v3, v4, v5, and v6 are the eigenvectors corresponding to the eigenvalues a1, a2, a3, a4, a5, and a6 respectively.

Since we don't have the eigenvectors explicitly given, we cannot calculate the exact value of a. However, we can say that a is equal