Matrix Addition

Property To add two matrices, $A$ and $B$, of the same dimensions, add each element in the first matrix to the element that is in the same location in the second matrix. $$ \begin{bmatrix} a {11} & a {12} \\ a {21} & a {22} \end{bmatrix} + \begin{bmatrix} b {11} & b {12} \\ b {21} & b {22} \end{bmatrix} = \begin{bmatrix} a {11} + b {11} & a {12} + b {12} \\ a {21} + b {21} & a {22} + b {22} \end{bmatrix} $$.

$$ \begin{bmatrix} 82 & 54 \\ 44 & 62 \end{bmatrix} + \begin{bmatrix} 91 & 46 \\ 22 & 45 \end{bmatrix} = \begin{bmatrix} 82+91 & 54+46 \\ 44+22 & 62+45 \end{bmatrix} = \begin{bmatrix} 173 & 100 \\ 66 & 107 \end{bmatrix} $$ $$ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix} $$ $$ \begin{bmatrix} 4 & 15 \\ 9 & 1 \end{bmatrix} + \begin{bmatrix} 4 & 15 \\ 9 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} $$.

Think of it like adding stats for two teams! To get the total, you just add the matching numbers from the same positions in each matrix. Just make sure the matrices are the same size, or it's like comparing apples to oranges—it won't work! This makes combining data sets super organized and quick.