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Matrix Addition

Lesson Contents

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 2×2 matrices looks like:

\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}

Rules of Matrix Addition

Matrix addition and matrix subtraction have the same rules. The operation for adding 2×3 matrices follows the same process as adding 2×2 matrices. The operation for adding 3×3 matrices follows the same process as well. The operation for adding 7×9 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 5×5, or must both be 3×6, etc). Make sure keep track of any negative numbers during matrix addition, they can get easily lost when working with a large matrix!

Example Problem

Here’s an example problem for 2×2 matrix addition.
Find the final matrix by adding the two matrices.
M1 = \begin{bmatrix} 5 & 8\\ 3 & 6 \end{bmatrix} and M2 = \begin{bmatrix} 6 & 2\\ 12 & 2 \end{bmatrix}

The matrices are both 2×2, so they meet the requirement of having the same dimension. Let’s add the two matrices by adding the numbers in like entry positions.
a1 + a2 = 5 + 6 = 11
b1 + b2 = 8 + 2 = 10
c1 + c2 = 3 + 12 = 15
d1 + d2 = 6 + 2 = 8
Now let’s plug the numbers into our final matrix.
M1 + M2 = \begin{bmatrix} 11 & 10\\ 15 & 8 \end{bmatrix}

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