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Row Addition

The final row operation you are allowed to do is row addition. This is basically just adding one row to another. The notation for row addition is: $R_i+R_j\longrightarrow R_j$. This means the $i^{th}$ row is added to the $j^{th}$ row and you replace the old $j^{th}$ row with the new summed one.

Example

Let's say the matrix is $\begin{bmatrix}a&b\\c&d\end{bmatrix}$.

Now let's perform the operation: $R_1+R_2\longrightarrow R_2$.

So now our matrix becomes $\begin{bmatrix}a&b\\\color{#8925AE}{a+c}&\color{#8925AE}{b+d}\end{bmatrix}$

But wait there's more!

Technically there are three row operations. But commonly row multiplication and addition are combined into one. This is just a quicker way that requires you to write less. You can always break them up separately if you don't feel comfortable. The notation would look like this: $cR_i+R_j\longrightarrow R_j$. This means the $i^{th}$ row is multiplied by a scalar $c$ and added to the $j^{th}$ row. This new summed $j^{th}$ row replaces the original $j^{th}$ row.

Example

Let's say once again the matrix is $\begin{bmatrix}a&b\\c&d\end{bmatrix}$.

Now let's perform the operation: $2R_2+R_1\longrightarrow R_1$.

So now our matrix becomes $\begin{bmatrix}\color{#8925AE}{a+2c}&\color{#8925AE}{b+2d}\\c&d\end{bmatrix}$

Notice that this is the same as performing $2R_2\longrightarrow R_2$ $\Bigg( \begin{bmatrix}a&b\\ \color{#29A603}{2c}&\color{#29A603}{2d}\end{bmatrix} \Bigg)$, then $R_2+R_1\longrightarrow R_1$ $\Bigg( \begin{bmatrix}\color{#8925AE}{a+}\color{#29A603}{2c}&\color{#8925AE}{b+}\color{#29A603}{2d}\\\color{#29A603}{2c}&\color{#29A603}{2d}\end{bmatrix} \Bigg)$ and then finally $\frac12\ R_2\longrightarrow R_2$ $\Bigg( \begin{bmatrix}\color{#8925AE}{a+}\color{#29A603}{2c}&\color{#8925AE}{b+}\color{#29A603}{2d}\\c&d\end{bmatrix} \Bigg)$

See how this gives you same result but we had to do a lot more work.

Matrix Row Operations: Row Addition

Our final type of matrix basic row operation is row addition! Row addition allows us to add two rows together, replacing an original row.

We will commonly see this technique used with systems of equations. When we solve a system, we can rewrite the coefficients are values in the matrix. Solving systems of equations is just one place we'd use this elementary row operation. You may also see it in gaussian elimination and with augmented matrices!

Example

Apply the row addition $R_2 + R_3 \rightarrow R_2$ to the following matrix:

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

We will leave the first and third rows alone, so we will just focus on the second row.

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

If we simplify,

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

(Video) Matrix Row Addition

by Anusha Rahman

In this video, Anusha Rahman goes over what matrix row addition and walks through an example.

Summary

With matrix row addition we are able to add one row of a matrix to another row of the matrix, replacing the old row in the matrix.

Matrix C: $\begin{bmatrix} 3&1&2 \\ 4&1&1 \\ 3&2&2 \\ \end{bmatrix}$

Let's say we want to add the first and the second row, to create a new second row. We can also think of this as: $R_1 + R_2 \rightarrow R_2$

$$\begin{bmatrix} 3&1&2 \\ 4&1&1 \\ 3&2&2 \\ \end{bmatrix} \rightarrow \begin{bmatrix} 3&1&2 \\ 4+3&1+1&1+2 \\ 3&2&2 \\ \end{bmatrix} \rightarrow \begin{bmatrix} 3&1&2 \\ 7&2&3 \\ 3&2&2 \\ \end{bmatrix}$$

We can see here that $R_1$ and $R_3$ did not change. We added $R_1$ to $R_2$, creating a new $R_2$.

Now, try the next example on your own!

Solution

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