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$$\textbf{Commutative Property of Matrix Addition}$$

Remember the commutative property of addition? As a reminder, it means that, given real numbers $a$ and $b,$ $$a+b=b+a$$

There is a similar commutative property when we are working with matrices! The Commutative Property of Matrix Addition states that, given $m\times n$ matrices $A$ and $B,$ $$A+B=B+A$$

(Important note: it is extremely important that these matrices are both $m\times n.$ If $A$ and $B$ are not the same size, you cannot add them.)

Let's look at a quick example! Consider the matrices $A= \begin{bmatrix} 0&3&1\\ 4&7&1 \end{bmatrix}$ and $B= \begin{bmatrix} 1&2&1\\ 1&3&2 \end{bmatrix}.$

Then: $$\begin{align} A+B &= \begin{bmatrix} 0&3&1\\ 4&7&1 \end{bmatrix}+\begin{bmatrix} 1&2&1\\ 1&3&2 \end{bmatrix}\\ &=\begin{bmatrix} 0+1 & 3+2 &1+1\\ 4+1 & 7+3 & 1+2 \end{bmatrix}\\ &=\begin{bmatrix} 1 & 5&2\\ 5& 10&3 \end{bmatrix} \end{align}$$

$$\begin{align} B+A &= \begin{bmatrix} 1&2&1\\ 1&3&2 \end{bmatrix}+\begin{bmatrix} 0&3&1\\ 4&7&1 \end{bmatrix}\\ &=\begin{bmatrix} 1+0 & 2+3&1+1\\ 1+4 & 3+7&2+1 \end{bmatrix}\\ &=\begin{bmatrix} 1 & 5&2\\ 5& 10&3 \end{bmatrix} \end{align}$$

So, $A+B=\begin{bmatrix} 1 & 5&2\\ 5& 10&3 \end{bmatrix}=B+A.$ Try it out the commutative property with matrices of your own!

$$\textbf{Associative Property of Matrix Addition}$$

Remember the associative property of addition? As a reminder, it means that, given real numbers $a, b,$ and $c,$ $$a+(b+c)=(a+b)+c$$

There is a similar associative property when we are working with matrix addition. The Associative Property of Matrix Addition states that, given $m\times n$ matrices $A, B,$ and $C,$ $$A+(B+C)=(A+B)+C$$

(Note: again, it is important that these matrices are all the same size, or else we cannot add them.)

Time for an example! Consider the matrices $A= \begin{bmatrix} 0&3\\ 4&7 \end{bmatrix}, B= \begin{bmatrix} 1&2\\ 1&3 \end{bmatrix},$ and $C=\begin{bmatrix} 0&1\\ 1&0 \end{bmatrix}.$

Then: $$\begin{align} A+(B+C) &= \begin{bmatrix} 0&3\\ 4&7 \end{bmatrix}+\Bigg(\begin{bmatrix} 1&2\\ 1&3 \end{bmatrix}+ \begin{bmatrix} 0&1\\ 1&0 \end{bmatrix}\Bigg)\\ &=\begin{bmatrix} 0&3\\ 4&7 \end{bmatrix}+\begin{bmatrix} 1+0 & 2+1\\ 1+1 & 3+0 \end{bmatrix}\\ &=\begin{bmatrix} 0&3\\ 4&7 \end{bmatrix}+\begin{bmatrix} 1 & 3\\ 2 & 3 \end{bmatrix}\\ &=\begin{bmatrix} 0+1 &3+3\\ 4+2 &7+3 \end{bmatrix}\\ &=\begin{bmatrix} 1 &6\\ 6 &10 \end{bmatrix} \end{align}$$

$$\begin{align} (A+B)+C &= \Bigg(\begin{bmatrix} 0&3\\ 4&7 \end{bmatrix}+\begin{bmatrix} 1&2\\ 1&3 \end{bmatrix}\Bigg)+ \begin{bmatrix} 0&1\\ 1&0 \end{bmatrix}\\ &=\begin{bmatrix} 0+1 & 3+2\\ 4+1 & 7+3 \end{bmatrix} + \begin{bmatrix} 0&1\\ 1&0 \end{bmatrix}\\ &=\begin{bmatrix} 1 & 5\\ 5 & 10 \end{bmatrix} + \begin{bmatrix} 0&1\\ 1&0 \end{bmatrix}\\ &=\begin{bmatrix} 1+0 & 5+1\\ 5+1 & 10+0 \end{bmatrix}\\ &=\begin{bmatrix} 1 & 6\\ 6& 10 \end{bmatrix} \end{align}$$

So, $A+(B+C)=\begin{bmatrix} 1 & 6\\ 6& 10 \end{bmatrix}=(A+B)+C.$ Try it out the associative property with matrices of your own!

$$\textbf{Identity Property of Matrix Addition}$$

The Identity Property of Matrix Addition states that, given an $m \times n$ matrix $A,$ there exists an $m \times n$ matrix $O$ such that: $$A+O=A$$

Here, $O$ is the $m\times n$ zero matrix! Let's do an example to clarify.

Consider the matrix $A= \begin{bmatrix} 0&1 & 0\\ 2&3 & 1\\ 1 & 0 & 1 \end{bmatrix}.$

If we let $O=\begin{bmatrix} 0&0&0\\ 0&0&0\\ 0&0&0 \end{bmatrix},$ then $$\begin{align} A+O&=\begin{bmatrix} 0&1&0\\ 2&3&1\\ 1&0&1 \end{bmatrix}+\begin{bmatrix} 0&0&0\\ 0&0&0\\ 0&0&0 \end{bmatrix}\\ &=\begin{bmatrix} 0+0&1+0&0+0\\ 2+0&3+0&1+0\\ 1+0&0+0&1+0 \end{bmatrix}\\ &=\begin{bmatrix} 0&1&0\\ 2&3&1\\ 1&0&1 \end{bmatrix}\\ &=A \end{align}$$

Basically, when you add a matrix and a zero matrix of the same size, you get back your original matrix!

$$\textbf{Inverse Property of Matrix Addition}$$

The Inverse Property of Matrix Addition states that, given an $m \times n$ matrix $A,$ there exists a unique matrix $-A$ such that: $$A+(-A)=O,$$ where $O$ is the $m \times n$ zero matrix.

Here's an example of how this works. Consider the matrix $A= \begin{bmatrix} 0&1\\ 2&3 \end{bmatrix}.$

Thus, $-A=-\begin{bmatrix} 0&1\\ 2&3 \end{bmatrix},$ and: $$\begin{align} A+(-A)&=\begin{bmatrix} 0&1\\ 2&3 \end{bmatrix}+\Bigg(-\begin{bmatrix} 0&1\\ 2&3 \end{bmatrix}\Bigg)\\ &=\begin{bmatrix} 0&1\\ 2&3 \end{bmatrix}-\begin{bmatrix} 0&1\\ 2&3 \end{bmatrix}\\ &=\begin{bmatrix} 0-0&1-1\\ 2-2&3-3 \end{bmatrix}\\ &=\begin{bmatrix} 0&0\\ 0&0 \end{bmatrix}\\ &=O, \text{ the $2 \times 2$ zero matrix!} \end{align}$$

In other words, to get the zero matrix, you can add any matrix with its negative counterpart!

(Video) Properties of Matrix Addition

by Anusha Rahman

In this video by Anusha Rahman, you can learn about the Commutative Property of Matrix Addition and the Associative Property of Matrix Addition.

Summary

The Commutative Property of Matrix Addition tells us that the order in which we add matrices doesn't matter. That's because the commutative property of addition applies for each entry addition in the two matrices.

$$F + G = H$$ $$G+F=H$$

Example

$$\begin{bmatrix} 3&2&3 \\ 4&9&1 \\ 10&1&4 \\ \end{bmatrix} + \begin{bmatrix} 1&3&1 \\ 5&7&3 \\ 1&12&5 \end{bmatrix} = \begin{bmatrix} 4&5&4 \\ 9&16&4 \\ 11&13&9 \\ \end{bmatrix}$$ $$\begin{bmatrix} 1&3&1 \\ 5&7&3 \\ 1&12&5 \\ \end{bmatrix} + \begin{bmatrix} 3&2&3 \\ 4&9&1 \\ 10&1&4 \end{bmatrix} = \begin{bmatrix} 4&5&4 \\ 9&16&4 \\ 11&13&9 \\ \end{bmatrix}$$

The Associative Property of Matrix Addition tells us that the order of parentheses when adding matrices doesn't matter.

$$D+(E+F) = (D+E)+F$$

Example

$$\begin{bmatrix} 2&4 \\ 3&5 \\ \end{bmatrix} + (\begin{bmatrix} 1&2 \\ 3&7 \\ \end{bmatrix} + \begin{bmatrix} 3&5 \\ 7&2 \\ \end{bmatrix}) = \begin{bmatrix} 2&4 \\ 3&5 \\ \end{bmatrix} + \begin{bmatrix} 4&7 \\ 10&9 \\ \end{bmatrix} = \begin{bmatrix} 6&11 \\ 13&14 \\ \end{bmatrix}$$ $$(\begin{bmatrix} 2&4 \\ 3&5 \\ \end{bmatrix} + \begin{bmatrix} 1&2 \\ 3&7 \\ \end{bmatrix}) + \begin{bmatrix} 3&5 \\ 7&2 \\ \end{bmatrix} = \begin{bmatrix} 3&6 \\ 6&12 \\ \end{bmatrix} + \begin{bmatrix} 3&5 \\ 7&2 \\ \end{bmatrix} = \begin{bmatrix} 6&11 \\ 13&14 \\ \end{bmatrix}$$

Properties of Matrix Addition

There are a couple of different properties that we need to keep in mind when working with matrices and matrix equations.

The Commutative Property of Matrix Addition

The Associative Property of Matrix Addition

When looking at the addition of matrices, the types of matrices need to be the same. That means that the matrix size (rows and columns) needs to be the same. For the above two properties, they must be matrices of the same order.

Going Further

Next up, you will also learn about scalar multiplication, which is basically the distributive property.

To learn more about the multiplicative identities of matrices, you can check out this lesson. You can learn about the property of multiplication.