Calculation of Eigenvalues of Matrix b
Matrix b can be expressed as:
b = a^4 - 3a^3 + 3a^2 - 2a + 8I

Step 1: Calculate a^2, a^3, and a^4
Given matrix a:
a = (1, -1; 2, 3)
a^2 = a * a = (1, -1; 2, 3) * (1, -1; 2, 3) = (1*1 + (-1)*2, 1*(-1) + (-1)*3; 2*1 + 3*2, 2*(-1) + 3*3) = (3, -4; 8, 7)
a^3 = a * a^2 = (1, -1; 2, 3) * (3, -4; 8, 7) = (1*3 + (-1)*8, 1*(-4) + (-1)*7; 2*3 + 3*8, 2*(-4) + 3*7) = (-5, -11; 30, 1)
a^4 = a * a^3 = (1, -1; 2, 3) * (-5, -11; 30, 1) = (1*(-5) + (-1)*30, 1*(-11) + (-1)*1; 2*(-5) + 3*30, 2*(-11) + 3*1) = (-35, -10; 55, -23)

Step 2: Substitute a^2, a^3, and a^4 back into b
b = (-35, -10; 55, -23) - 3(-5, -11; 30, 1) + 3(3, -4; 8, 7) - 2(1, -1; 2, 3) + 8I
Simplify b to get the final expression in terms of a.

Step 3: Find Eigenvalues of Matrix b
Once you have the final expression for b, you can find the eigenvalues by solving the characteristic equation det(b - λI) = 0, where λ is the eigenvalue.
Solve the characteristic equation to find the eigenvalues of matrix b.
In this way, you can calculate the eigenvalues of matrix b using the given matrix a and the expression for b.