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[SOLVED] Define matrix to get a row operation of type 1

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,028
Hey!! :eek:

We have the matrices \begin{equation*}a=\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{pmatrix}, \ \ E_{1,3}=\begin{pmatrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}, \ \ u_n=\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &0 & 1\end{pmatrix}\end{equation*}

I have calculated:
\begin{align*}\left (u_n+2E_{1,3}\right )a&=\left (\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &0 & 1\end{pmatrix}+2\begin{pmatrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}\right )\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{pmatrix} \\ &=\left (\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &0 & 1\end{pmatrix}+\begin{pmatrix}0 & 0 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}\right )\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{pmatrix} \\ &= \begin{pmatrix}1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 &0 & 1\end{pmatrix}\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{pmatrix} \\ &= \begin{pmatrix}15 & 18 & 21 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{pmatrix}\end{align*}

Let $a \in \mathbb{R}^{n\times m}$. Determine a matrix $b \in \mathbb{R}^{n\times n}$, such that the product $ba$ is a row operation of type 1.

The row operatioon of type 1 is that we multiply one row by a scalar and add it to another row. For that do we define $b$ to be as above, i.e. in the form $\left (u_n+sE_{i,j}\right )$ ? (Wondering)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,711
Hey mathmari !!

Yep.
In the example we saw that we added 2 times row 3 to the first row.
So now we generalize to s times row j that we add to row i. (Thinking)
 

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,028