Chapter 1, Problem 13CTA

This chapter test consists of true−or−false questions. In each case, answer true if the statement is always true and false otherwise. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true. For example, consider the following statements about $n\times n$ matrices A and B:

1. $A+B=B+A$

$AB=BA$

Statement (i) is always true. Explanation: The $\left(i,j\right)$ entry of $A+B$ is $a_{ij}+b_{ij}$ and the $\left(i,j\right)$ entry of $B+A$
is $b_{ij}+a_{ij}$ . Since $a_{ij}+b_{ij}=b_{ij}+a_{ij}$ for each i and j, it follows that $A+B=B+A$ .
The answer to statement (ii) is false. Although the statement may be true in some cases, it is not always true. To show this, we need only exhibit one instance in which equality fails to hold. For example, if

$A=\left[\begin{array}{cc}1& 2\\ 3& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}2& 3\\ 1& 1\end{array}\right]$

Then

$AB=\left[\begin{array}{cc}4& 5\\ 7& 10\end{array}\right]$ and $BA=\left[\begin{array}{cc}11& 7\\ 4& 3\end{array}\right]$

This proves that statement (ii) is false.

13. If E is an elementary matrix, then $E^T$ is also an elementary matrix.