行乘法

Row Multiplication

Row multiplication is also a very simple row operation. You can take any row and multiply all its elements by any constant. It is denoted by $cR_i\longrightarrow R_i$ which means you're multiplying the $i^{th}$ row by $c$ and replacing the old row with this new multiplied one.

Example

Let $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$

Now let's perform the operation: $2R_2\longrightarrow R_2$

Now our matrix becomes: $\begin{bmatrix}a&b\\\color{red}{2c}&\color{red}{2d}\end{bmatrix}$

Matrix Row Operations: Row Multiplication

The next type of matrix row operations is row multiplication. Here, we multiply each term in a given row by a scalar value.

We see this concept used more commonly in linear algebra (dot product) and computer science.

We multiply each term in a given row by a scalar value. This is NOT when we multiply two matrices.

Another thing to keep in mind is that because it is row multiplication, we do not multiply by the columns of the matrix. Mixing up the rows and columns is a tricky mistake, so keep an eye out!

Example

Apply the row operation $R_1 \rightarrow 3R_1$ to the following matrix:

$$\begin{bmatrix} 2&1&0 \\ 3&4&2 \\ 3&6&1 \\ \end{bmatrix}$$

The number of rows is $3$, and the number of columns is $3$.

To do that, we need to multiply each term in the first row by $3$. We are multiplying a row.

$$\begin{bmatrix} 3*2&3*1&3*0 \\ 3&4&2 \\ 3&6&1 \\ \end{bmatrix}$$

If we simplify, we get:

$$\begin{bmatrix} 6&3&0 \\ 3&4&2 \\ 3&6&1 \\ \end{bmatrix}$$

This is the result matrix.

We will use this technique when solving systems of linear equations, so be sure to practice well.

(Video) Matrix Row Multiplication

by Anusha Rahman

In this video, Anusha Rahman goes over what row multiplication is and goes through a couple of examples.

Summary

Row multiplication is when we multiply each entry by a constant value, thereby replacing the old entries with the new entries.

$$cR_x \rightarrow R_x$$

Example:

$2R_2 \rightarrow R_2$

$\begin{bmatrix} 2&5&7 \\ 1&3&2 \\ 0&1&3 \\ \end{bmatrix} \rightarrow \begin{bmatrix} 2&5&7 \\ 2\cdot 1 &2\cdot 3&2\cdot 2 \\ 0&1&3 \\ \end{bmatrix} \rightarrow \begin{bmatrix} 2&5&7 \\ 2&6&4 \\ 0&1&3 \\ \end{bmatrix}$

The middle matrix is the intermediate matrix, which allows us to see the intermediate step of multiplying each entry in the second row by the constant value, $2$.

Try the next example on your own!

$3R_2 \rightarrow R_2$ for the following matrix: $\begin{bmatrix} 1&7 \\ 9&3 \\ \end{bmatrix}$

$\begin{bmatrix} 1&7 \\ 9&3 \\ \end{bmatrix} \rightarrow \begin{bmatrix} 1&7 \\ 3\cdot 9 & 3\cdot 3 \\ \end{bmatrix} \rightarrow \begin{bmatrix} 1&7 \\ 27&9 \\ \end{bmatrix}$