পুনরালোচনা: ম্যাট্রিক্সের যোগ ও বিয়োগ

Subtracting Matrices

The $\boxed{\text{first rule}}$ in subtracting matrices together is that they must have the same dimensions. This means that the matrices must be the same size. Look at the row and column.

Take a look at what we mean by that: $$\text{**Nope.** We *cannot* subtract these two matrices from each other}$$ $$\text{because they have **different dimensions**.}$$ $$$$ $$\text{**Looks good.** We *can* subtract these two matrices from each other}$$ $$\text{because they have **the same dimensions**.}$$

The $\boxed{\text{second rule}}$ is that matrices are subtracted entry-by-entry. This means that to subtract two matrices from each other, you need to subtract the numbers in each matrix that are in the same position. These numbers are called entries.

Because we are performing subtraction, we have to pay attention to the order of our matrices. Subtracting the first matrix from the second will NOT produce the same results as subtracting the second matrix from the first.

Let's look at an example:

Remember, because this is subtraction, the order in which we subtract matters. For example, we need to do $4-3$ to get the top left entry in our new matrix. We cannot do $3-4$, since this would give us a different answer.

Adding Matrices

The first rule in adding matrices together is that they must have the same dimensions. This means that the matrices must be the same size.

Take a look at what we mean by that:

$$\begin{pmatrix} 3 & 1 \\ 2 & 1 \end{pmatrix} + \begin{pmatrix} 4 & 6 & 5\\ 0 & 2 & 9 \end{pmatrix}$$ $$\text{**No ma'am.** We *cannot* add these two matrices}$$ $$\text{because they have **different dimensions.**}$$ $$\begin{pmatrix} 3 & 1 \\ 2 & 1 \end{pmatrix} + \begin{pmatrix} 0 & 5\\ 6 & 6 \end{pmatrix}$$ $$\text{**Much better.** We *can* add these two matrices}$$ $$\text{because they have **the same dimensions.**}$$

$$\begin{pmatrix} -7 & 5 \\ 3 & 0 \end{pmatrix}, \begin{pmatrix} 4 & 10 \\ 2 & 5 \\ 1 & -1 \end{pmatrix}$$

$$\begin{pmatrix} 3 & -1 & 6 \\ 2 & 0 & 2 \\ 0 & 4 & 4 \end{pmatrix}, \begin{pmatrix} 0 & 2 & 9\\ -1 & 2 & -3 \\ 5 & 7 & 3 \end{pmatrix}$$

The second rule to adding matrices is that they are added entry by entry. This means that, to add two matrices together, you need to add together the numbers in each matrix that are in the same position. (These numbers are called entries.)

Because we are performing addition, it does not matter if we add the first matrix to the second one or if we add the second matrix to the first one. It will all produce the same answer. This property is called commutativity.

Let's look at a few examples:

$$\begin{pmatrix} -1 & 0 & 3 \\ 4 & -2 & 8 \end{pmatrix}+ \begin{pmatrix} 4 & 6 & 5\\ 0 & -2 & 9 \end{pmatrix}$$ $$\text{These matrices are both }3\text{-by-}2\text{.}\\ \text{Since they have the same dimensions, we can add them together.}$$ $$=\begin{pmatrix} -1+4 & 0+6 & 3+5 \\ 4+0 & -2-2 & 8+9 \\ \end{pmatrix}$$ $$=\begin{pmatrix} 3 & 6 & 8 \\ 4 & -4 & 17 \\ \end{pmatrix}$$

Notice how, in the second step, we are adding the numbers together that are in the same position in each matrix.

Let's look at one more example that's formatted just a bit differently. Here it is: $$A=\begin{pmatrix} 3 & 5 \\ 2 & 6 \\ \end{pmatrix} \quad B=\begin{pmatrix} 4 & -1 \\ 2 & 0 \\ \end{pmatrix} \quad C=\begin{pmatrix} 7 & 3 & 2 \\ 0 & 0 & 5 \\ \end{pmatrix}$$

1. $A+B=\begin{pmatrix} 7 & 4 \\ 4 & 6 \\ \end{pmatrix}$
2. $A+C=\begin{pmatrix} 10 & 8 & 2 \\ 2 & 6 & 5 \\ \end{pmatrix}$
3. $B+C=\begin{pmatrix} 11 & 2 \\ 2 & 0 \\ \end{pmatrix}$

To figure out the answer here, we need to remember our two rules for adding matrices:

1. The matrices must have the same dimensions.
2. Add the matrices entry by entry.

We can see that the only two matrices that are the same size are $A$ and $B$. This means that we CANNOT add $A$ and $C$ and we CANNOT add $B$ and $C$. We are only able to add together $A$ and $B$, so we only need to check the first option: $$A+B=\begin{pmatrix} 3 & 5 \\ 2 & 6 \\ \end{pmatrix}+ \begin{pmatrix} 4 & -1 \\ 2 & 0 \\ \end{pmatrix}= \begin{pmatrix} 3+4 & 5-1 \\ 2+2 & 6+0 \\ \end{pmatrix}= \begin{pmatrix} 7 & 4 \\ 4 & 6 \\ \end{pmatrix}$$ So option 1 is correct.

Adding and Subtracting Matrices

When you add or subtract matrices, there are two basic rules we need to always keep in mind.

Rule #1: Same Dimensions

In order for two matrices to be added or subtracted, they need to have the same dimensions. So for example, you can only add $3x3$ matrices to $3x3$ matrices. These matrices have the same dimensions.

Rule #2: Add and Subtract Corresponding Values

Matrix addition and subtraction requires entry-by-entry adding or subtracting. So you add or subtract the value in the same respective position of each matrix. For addition, the order of the matrices doesn't matter.

In the Future

Next steps will include learning scalar multiplication!

(Video) Adding Matrices

by Math Meeting

This video by Math Meeting shows how to add matrices.

The biggest thing to note with matrix addition is that the matrices have to have the same dimensions, a.k.a they have to be the same size. For example, a $2 \times 2$ matrix can only be added with other $2 \times 2$ matrices. The resulting matrix will then also be a $2 \times 2$ matrix.

Another key concept to note is that when adding matrices with the same size, you must add the corresponding numbers together. For example, in the $2 \times 2$ matrix addition, the values in the top left corner must be added together, and so on for the other corners.

Matrix Subtraction: Example

Subtracting matrices is actually very similar to adding matrices.

Recall that we can only add one matrix to another if they both have the same dimensions; the same is true for subtraction.

Here is an example:

The number of columns is $2$, and the number of rows is $2$.

Option 1: Transform Subtraction to Addition

Option 2: Subtract Directly