Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have two elementary row operation matrices (elimination matrices):

$E_{31} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{bmatrix} $ (adds row $1$ to row $3$)

$E_{13} = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} $ (adds row $3$ to row $1$)

Am I correct in saying that if I first want $E_{31}$ applied and second want $E_{13}$ applied to a matrix, it is written as $E_{13}E_{31}$? I think of it as $E_{13}(E_{31}M)$.

My homework question asks what $3$ by $3$ matrix adds row $1$ to row $3$ and then adds row $3$ to row $1$:

Question

I thought the answer was $E_{13}E_{31}$; that is, $E_{31}$ (adds row 1 to row 3) gets applied first, and then $E_{13}$ (adds row 3 to row 1) gets applied after.

The book's answer is:

Answer

You can get the book's answer by multiplying $E_{13}$ with the matrix $E_{31}$.

Why did I get the wrong answer? I have used this same logic to correctly answer previous questions involving the order of multiplying two EROs.


This question is $\#10$ in Section 2.2 of Introduction to Linear Algebra by Gilbert Strang, 4th edition. According to the answer below, the book's solution has a misprint. The matrix presented isn't wrong, but the two E matrices multiplied to make that step is wrong.

share|improve this question
    
Yeah, something like that. I added the (adds row 1 to row 3) to point out that the subscripts are reversed. Like, $E_{xy}$ means first y and then x. –  Jason Jan 24 '13 at 23:13
    
@RustynYazdanpour He still got the other answers right. I guess it's most likely that the book has this one wrong. –  Git Gud Jan 24 '13 at 23:13
    
@Jason Why don't you try multiplying your matrix by an arbitrary $3\times 3$ matrix and check who's right? –  Git Gud Jan 24 '13 at 23:14
    
You mean multiplying $E_{13}E_{31}$ (my answer) with some random 3 by 3 matrix $M$? –  Jason Jan 24 '13 at 23:16
    
@Jason Yes, that's what I mean. –  Git Gud Jan 24 '13 at 23:17
show 5 more comments

1 Answer 1

up vote 2 down vote accepted

The correct answer is $E_{13}E_{31}= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}=P$ (say). You said that the book's answer is $E_{31}E_{13}=P$. If the book literally says that, then it is correct in that $P$ is the answer, but it is also wrong because $P=E_{13}E_{31}$, not $P=E_{31}E_{13}$. Since I don't have the book at hand, it's hard to say if the book is wrong or you have quoted the book wrongly.

share|improve this answer
    
The line that starts with "The book's answer is", the words and matrix after that is exactly pixel for pixel what the solutions has. No other letters before or after. –  Jason Jan 24 '13 at 23:43
    
Just kidding, it says "Test on the identity matrix!" after that. But does that mean anything? –  Jason Jan 24 '13 at 23:44
    
@Jason Then as I said, it is correct that $P$ is the answer, but it is wrong that $E_{31}E_{13}=P$. As to testing on $I_3$, he means if the two elementary matrices are multiplied in the opposite order, you will get a wrong result that is easy to spot. –  user1551 Jan 24 '13 at 23:47
    
What is "P(say)"? By the way, I've uploaded images of the book's question and answer. –  Jason Jan 24 '13 at 23:50
    
@Jason It means "let us call this 3-by-3 matrix $P$". –  user1551 Jan 24 '13 at 23:51
show 3 more comments

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.