If you find any mistakes, please make a comment! Thank you.

Definition and properties of matrices with a single nonzero entry

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.6


(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 $p \neq i$, then $e_{p,k} = 0$, so that $c_{p,q} = 0$. If $p = i$, then $c_{p,q} = a_{j,q}$; thus the $i$-th row of $E_{i,j}A$ is the $j$-th row of $A$, and all other entries are 0.

(2) The proof for $AE_{i,j}$ is very similar.

(3) By the above arguments, $E_{p,q}A$ is the matrix whose $p$-th row is the $q$-th row of $A$, and all other entries are $0$. Then $E_{p,q}AE_{r,s}$ is the matrix whose $s$-th column is the $r$-th column of $E_{p,q}A$, which is all zeroes except for the $p$-th row, whose entry is the $(q,r)$ entry of $A$, and all other entries are zero.


This website is supposed to help you study Linear Algebras. Please only read these solutions after thinking about the problems carefully. Do not just copy these solutions.
Close Menu