Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.1 Solution: In Exercise 7.2.6, we saw that $AE_{i,j}$ is the matrix whose j-th column is the…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.31 Solution: We begin with a lemma. Lemma: Let $R$ be a ring with $1 \neq 0$.…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.21 Solution: First, let $I \subseteq M_n(R)$ be a two-sided ideal. Let $J \subseteq R$ consist of…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.14 Solution: Define $\varphi : \mathbb{H} \rightarrow M_4(\mathbb{R})$ as follows. $$a+bi+cj+dk \mapsto \begin{bmatrix} a & b &…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.13 Solution: Define $$\varphi : \mathbb{C} \rightarrow M_2(\mathbb{R}) by a+bi \mapsto \begin{bmatrix} a & b \\ -b…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.12 Solution: We begin with a lemma. Lemma: If $D \in \mathbb{Z}$ is not a perfect square,…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.7 Solution: Let $A,B \in R$ be arbitrary, with $A = \begin{bmatrix} a_1 & b_1 \\ 0…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.7 Let $R$ be a commutative ring with 1. Prove that the center of the ring $M_n(R)$…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.6 Solution: (1) By definition, $E_{i,j}A = [c_{p,q}]$, where $$c_{p,q} = \sum_{k=1}^n e_{p,k}a_{k,q}.$$ Note that if $p…