How to Add Matrices

To add two matrices, we simply add the numbers that are in like entry positions. Matrices must have the same dimension to be addable.

Here is what matrix addition for 2x2 matrices looks like:

$$\begin{align}& \begin{bmatrix} a1 & b1\\ c1 & d1 \end{bmatrix} + \begin{bmatrix} a2 & b2\\ c2 & d2 \end{bmatrix} = \begin{bmatrix} a1 + a2 & b1 + b2\\ c1 + c2 & d1 + d2 \end{bmatrix} \end{align}$$

Rules of Matrix Addition

Matrix addition and matrix subtraction have the same rules. The operation for adding 2x3 matrices follows the same process as adding 2x2 matrices. The operation for adding 3x3 matrices follows the same process as well. The operation for adding 7x9 matrices also follows the same process.

In fact, we can add matrices of any size as long as the two matrices being added have the same dimension (i.e. must both be 5x5, or must both be 3x6, etc). Make sure keep track of any negative numbers during matrix addition, they can get easily lost when working with a large matrix!

Matrix Addition Example Problem

Let's work through an example problem together to reinforce our skills with adding matrices.

Find the final matrix by adding the two matrices:

$$\begin{align}& \text{M1 } = \begin{bmatrix} 5 & 8\\ 3 & 6 \end{bmatrix} \hspace{3ex} \text{ M2 } = \begin{bmatrix} 6 & 2\\ 12 & 2 \end{bmatrix} \end{align}$$

Solution:

1. The matrices are both 2x2, so they meet the requirement of having the same dimension. Let's add the two matrices by adding the numbers in like entry positions.
2. a1 + a2 = 5 + 6 = 11
3. b1 + b2 = 8 + 2 = 10
4. c1 + c2 = 3 + 12 = 15
5. d1 + d2 = 6 + 2 = 8
6. Now let's plug the numbers into our final matrix.

$$\begin{align}& \text{M1 } + \text{ M2 } = \begin{bmatrix} 11 & 10\\ 15 & 8 \end{bmatrix} \end{align}$$