# Thread: Matrix as product of elementary matrices

1. ## Matrix as product of elementary matrices

Hey guys I need to express this matrix as the product as elementary matrices

$\displaystyle \begin{pmatrix} 1 & 0 & -2 \\ 0 & 4 & 3 \\ 0 & 0 & 1 \end{pmatrix}$

I get this answer but the book has a different answer can someone tell me where I went wrong.

$\displaystyle \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac1{4} & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & \frac3{4} \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

Thanks

2. Originally Posted by evant8950
Hey guys I need to express this matrix as the product as elementary matrices

$\displaystyle \begin{pmatrix} 1 & 0 & -2 \\ 0 & 4 & 3 \\ 0 & 0 & 1 \end{pmatrix}$

I get this answer but the book has a different answer can someone tell me where I went wrong.

$\displaystyle \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac1{4} & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & \frac3{4} \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

Thanks
If you multiple your answer out, you will see it doesn't yield the original matrix.

3. The book has

$\displaystyle \displaystyle\begin{bmatrix}1&0&0\\0&4&0\\0&0&1\en d{bmatrix}\cdot\begin{bmatrix}1&0&0\\0&1&\frac{3}{ 4}\\0&0&1\end{bmatrix}\cdot\begin{bmatrix}1&0&-2\\0&1&0\\0&0&1\end{bmatrix}$

Correct?

4. No the book has:
$\displaystyle \displaystyle\begin{bmatrix}1&0&0\\0&4&0\\0&0&1\en d{bmatrix}\cdot\begin{bmatrix}1&0&0\\0&1&3\\0&0&1\ end{bmatrix}\cdot\begin{bmatrix}1&0&-2\\0&1&0\\0&0&1\end{bmatrix}$

5. Originally Posted by evant8950
No the book has:
$\displaystyle \displaystyle\begin{bmatrix}1&0&0\\0&4&0\\0&0&1\en d{bmatrix}\cdot\begin{bmatrix}1&0&0\\0&1&3\\0&0&1\ end{bmatrix}\cdot\begin{bmatrix}1&0&-2\\0&1&0\\0&0&1\end{bmatrix}$
Multiplying that out doesn't yield the original matrix though.

6. I'm sorry the book has the solutions in this order:

$\displaystyle \displaystyle\begin{bmatrix}1&0&-2\\0&1&0\\0&0&1\end{bmatrix}\cdot\begin{bmatrix}1& 0&0\\0&1&3\\0&0&1\end{bmatrix}\cdot\begin{bmatrix} 1&0&0\\0&4&0\\0&0&1\end{bmatrix}$

I multiplied this out and it gave the right solution.

7. Originally Posted by evant8950
I'm sorry the book has the solutions in this order:

$\displaystyle \displaystyle\begin{bmatrix}1&0&-2\\0&1&0\\0&0&1\end{bmatrix}\cdot\begin{bmatrix}1& 0&0\\0&1&3\\0&0&1\end{bmatrix}\cdot\begin{bmatrix} 1&0&0\\0&4&0\\0&0&1\end{bmatrix}$

I multiplied this out and it gave the right solution.
When you doing the elementary row operations, you would think 2 times row 3 plus row 1 would work, but think of it as the value minus your operation. Therefore, you have -2-(-2) to obtain 0.

From the above explantion, you will see we have 3-3 which is 0 so good.

And 4*1 gives us the 4.

Do you understand now?

8. Originally Posted by evant8950
Hey guys I need to express this matrix as the product as elementary matrices

$\displaystyle \begin{pmatrix} 1 & 0 & -2 \\ 0 & 4 & 3 \\ 0 & 0 & 1 \end{pmatrix}$

I get this answer but the book has a different answer can someone tell me where I went wrong.

$\displaystyle \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac1{4} & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & \frac3{4} \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

Thanks
The "KEY" idea is that multipying by an elemtry matrix is just like doing a row operation.

$\displaystyle \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0& 1\end{bmatrix}$

The first row op would be to multiply row 2 by 4 this gives

$\displaystyle E_1=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0& 1\end{bmatrix}$

Now you want to take $\displaystyle 3R_3+R_2=R_2$ This gives

$\displaystyle E_2=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 3 \\ 0 & 0& 1\end{bmatrix}$

and finally for the last one

$\displaystyle E_3=\begin{bmatrix} 1 & 0 & -2 \\ 0 & 1 & 0 \\ 0 & 0& 1\end{bmatrix}$

9. How do i get 1 where 4 is if I don't multiple by 1/4?

Thanks

10. Originally Posted by evant8950
How do i get 1 where 4 is if I don't multiple by 1/4?

Thanks
You need to get a 4 there. What times 1 equals 4?

11. So I am doing the inverse row operation to the identity matrix of what I would do as a row operation on the matrix itself?

12. Originally Posted by evant8950
So I am doing the inverse row operation to the identity matrix of what I would do as a row operation on the matrix itself?
You are doing a LU Factorization.

13. I think I am confused. Am I using elementary matrices to get A in reduced row echelon form? Then multiplying the elementary matrices to get A?

Thanks for the help

14. Originally Posted by evant8950
I think I am confused. Am I using elementary matrices to get A in reduced row echelon form? Then multiplying the elementary matrices to get A?

Thanks for the help
If A is nonsingular, then A is row equivalent to I.

$\displaystyle A=E_kE_{k-1}\cdots E_1I$