#### In this paper, we will determine the minimal and** characteristic polynomials of symmetric matrices** with entries in a field of characteristic two. More precisely, we will prove the following results. Theorem 1.1 Let k be a field of** characteristic** two, and let f ∈ k [ X] be a monic polynomial of degree n ≥ 1. Find the **characteristic polynomial** and the eigenvalues of the **matrices** in Exercises 1–8. 4 . [ 5 − 3 − 4 3 ] Textbook Question. Chapter 5.2, Problem 4E. ... In Problems 1 -6, express the given system of differential equations in **matrix** notation. x=7x+2y,y=3x2y. The roots are: λ = −4 ±√−36 2 λ = - 4 ± - 36 2. We see that the √−36 - 36 is equal to 6i, such that the eigenvalues become: λ = −4 ±6i 2 = − 2± 3i λ = - 4 ± 6 i 2 = - 2 ± 3 i. 2x2 **Matrix** Calculators : To compute the **Characteristic Polynomial** of a 3x3 **matrix**, CLICK HERE. To compute the Trace of a 2x2 **Matrix**, CLICK HERE.

**symmetric**

**polynomial**is a

**polynomial**P(X 1, X 2, , X n) in n variables, such that if any of the variables are interchanged, one obtains the same

**polynomial**. Formally, P is a

**symmetric**

**polynomial**if for any permutation σ of the subscripts 1, 2, ..., n one has P(X σ(1), X σ(2), , X σ(n)) = P(X 1, X 2, , X n).

**Symmetric**

**polynomials**arise naturally in the study of. Recall that if Ais

**symmetric**then all eigenvalues of Aare real. Therefore, if Ais

**symmetric**with eigenvalues 1;:::; n, the

**characteristic polynomial**is a real-rooted

**polynomial**with roots 1;:::; nand we can write, ˜[A](t) = (t 1)(t 2):::(t n) Note that by de nition the

**characteristic polynomial**is invariant under rotation. That is, it only. $ \def\P{\mathsf{\sf P}} \def\E{\mathsf{\sf E}} \def\Var{\mathsf{\sf Var}} \def\Cov{\mathsf{\sf Cov}} \def\std{\mathsf{\sf std}} \def\Cor{\mathsf{\sf Cor}} \def\R.