# Define matrix to get a row operation of type 1

• MHB
• mathmari
In summary, the conversation discussed matrices and their operations, specifically the row operation of type 1 where a row is multiplied by a scalar and added to another row. The example given showed how to calculate such an operation and the conversation went on to discuss generalizing this operation.
mathmari
Gold Member
MHB
Hey!

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)

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)

Klaas van Aarsen said:
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)

Ok! Thank you! (Sun)

## What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is commonly used in mathematics, statistics, and other scientific fields to represent and manipulate data.

## What is a row operation of type 1 on a matrix?

A row operation of type 1 on a matrix is a transformation that involves swapping the positions of two rows in the matrix. This can be done by interchanging the corresponding elements in each row.

## Why is a row operation of type 1 important?

A row operation of type 1 is important because it allows us to manipulate a matrix in a way that can make it easier to solve equations, find inverses, or perform other operations. It also helps to simplify and organize the data in the matrix.

## How do you define a matrix for a row operation of type 1?

To define a matrix for a row operation of type 1, you need to specify the two rows that will be swapped. This can be done by indicating the row numbers or by writing out the rows themselves. For example, if you want to swap the first and third rows, you would write R1 ↔ R3.

## Can a row operation of type 1 change the properties of a matrix?

No, a row operation of type 1 does not change the properties of a matrix. It only changes the order of the rows, but the values and dimensions of the matrix remain the same. This means that the determinant, rank, and other properties of the matrix will not be affected by a row operation of type 1.

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