Concept# Matrix addition

Summary

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.
For a vector, \vec{v}!, adding two matrices would have the geometric effect of applying each matrix transformation separately onto \vec{v}!, then adding the transformed vectors.
:\mathbf{A}\vec{v} + \mathbf{B}\vec{v} = (\mathbf{A} + \mathbf{B})\vec{v}!
However, there are other operations that could also be considered addition for matrices, such as the direct sum and the Kronecker sum.
Entrywise sum
Two matrices must have an equal number of rows and columns to be added. In which case, the sum of two matrices A and B will be a matrix which has the same number of rows and columns as A and B. The sum of A and B, denoted A + B, is computed by adding corresponding elements of A and B:
:\begin{align}
\mathbf{A}+\mathbf{B} & = \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \
a_{2

Official source

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