{%hackmd 5xqeIJ7VRCGBfLtfMi0_IQ %}

# Find the characteristic polynomial

## Problem

Run the SageMath code below or simply click [here](https://sagecell.sagemath.org/?q=zziwzi).

load("https://raw.githubusercontent.com/jephianlin/LA-notebook/main/lingeo.py")

### code
set_random_seed(None)

n = 3
A = matrix(n, random_int_list(### code
set_random_seed(None)

n = 3
A = matrix(n, random_int_list(n**2, 2))

for k in range(n + 1):
    print("k =", k)
    if k == 0:
        print("s0 = 1")
    else:
        sk = 0
        print("alpha det(A[alpha])")
        for alpha in Combinations(list(range(n)), k):
            detAalpha = A[alpha,alpha].determinant()
            sk += detAalpha
            print(alpha, detAalpha)
        print("s%s ="%k, sk)

print("characteristic polynomial:", (-1)**n * A.charpoly())n**2, 2))
print("A =")
print(A)
print()

for k in range(n + 1):
    print("k =", k)
    if k == 0:
        print("s0 = 1")
        print()
    else:
        sk = 0
        print("alpha det(A[alpha])")
        for alpha in Combinations(list(range(n)), k):
            detAalpha = A[alpha,alpha].determinant()
            sk += detAalpha
            print(alpha, detAalpha)
        print("s%s ="%k, sk)
        print()

print("characteristic polynomial:", (-1)**n * A.charpoly())

Replace `None` with your favorite number and run the code again. The outcome is a matrix $A$. Find the characteristic polynomial of $A$ and check your answer with the output.

*This note can be found at Course website > Learning resources.*