Matrix Addition

Okay, we're good with addition! Now let's say we had a bunch of numbers in an array, or as we will call it, a matrix, like so: $$\begin {pmatrix} 3 & 4 & 0 \\ 1 & -1 & 2 \\ 4 & 5 & -6 \end {pmatrix}$$And let's say we had another one! $$\begin {pmatrix} 4 & 5 & -3 \\ 10 & -9 & 1 \\ 0 & -1 & 7 \end {pmatrix}$$These two matrices are just numbers arranged in a nice format. But they are still numbers! Math is nice, so what it tells us is that, if we wanted to "add" these two matrices, we could treat them the same way we would treat numbers. In other words, we could just add the individual elements: $$\begin {align*} \begin {pmatrix} 3 & 4 & 0 \\ 1 & -1 & 2 \\ 4 & 5 & -6 \end {pmatrix} + \begin {pmatrix} 4 & 5 & -3 \\ 10 & -9 & 1 \\ 0 & -1 & 7 \end {pmatrix} &= \begin {pmatrix} 3 + 4 & 4 + 5 & 0 - 3 \\ 1 + 10 & -1 - 9 & 2 + 1 \\ 4 + 0 & 5 - 1 & -6 + 7 \end {pmatrix} \\&= \begin {pmatrix} 7 & 9 & -3 \\ 11 & -10 & 3 \\ 4 & 4 & 1 \end {pmatrix} \end {align*}.$$Wasn't that easy? Just add the corresponding elements in the matrices!

Right! Adding matrices is illegal when the sizes are not the same. We can use matrix addition to add vectors $-$ they are just like coordinates. Matrices can also be used in many other contexts; e.g. linear algebra. Be sure you understand how to add matrices correctly you move on the more advanced operations.