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In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.
For a Vector space, , adding two matrices would have the geometric effect of applying each matrix transformation separately onto , then adding the transformed vectors.
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{bmatrix} +
\begin{bmatrix}
b_{11} & b_{12} & \cdots & b_{1n} \\
b_{21} & b_{22} & \cdots & b_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
b_{m1} & b_{m2} & \cdots & b_{mn} \\
\end{bmatrix} \\
& = \begin{bmatrix}
a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\
a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \\
\end{bmatrix} \\
\end{align}\,\! Or more concisely (assuming that ):
For example:
\begin{bmatrix}
1 & 3 \\
1 & 0 \\
1 & 2
\end{bmatrix}
+
\begin{bmatrix}
0 & 0 \\
7 & 5 \\
2 & 1
\end{bmatrix}
=
\begin{bmatrix}
1+0 & 3+0 \\
1+7 & 0+5 \\
1+2 & 2+1
\end{bmatrix}
=
\begin{bmatrix}
1 & 3 \\
8 & 5 \\
3 & 3
\end{bmatrix}
Similarly, it is also possible to subtract one matrix from another, as long as they have the same dimensions. The difference of A and B, denoted , is computed by subtracting elements of B from corresponding elements of A, and has the same dimensions as A and B. For example:
1 & 3 \\
1 & 0 \\
1 & 2
\end{bmatrix}
-
\begin{bmatrix}
0 & 0 \\
7 & 5 \\
2 & 1
\end{bmatrix}
=
\begin{bmatrix}
1-0 & 3-0 \\
1-7 & 0-5 \\
1-2 & 2-1
\end{bmatrix}
=
\begin{bmatrix}
1 & 3 \\
-6 & -5 \\
-1 & 1
\end{bmatrix}
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