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 An m x n (m by n) matrix A is a fuction (function) whose domain is the set of pairs (i,j) were 1<= i <= m and 1 <=j <= n and his range is a scalar field commonly denoted by the letter F ( usually the real numbers or the complex nu​mbers ). Seen a matrix as a fuction (function) may be hard to imagine thats why we write them as a rectangle with m rows and n columns, inside of wich the element ubicated in the i row and the j column is the value of the function A(i,j).
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If we have a matrix A whose size is m x n (m by n) and we have and (a) scalar k that belongs to the scalar field F, then we define de (the) product between A and k by (kA)(i,j) = k(A(i,j), that is the element (i,j) from the matrix (kA), in other words, is the element (i,j) of the matrix A multiply (multiplied) by the scalar K

If we have two matrices A,B, both of size m x n (m by n) , then we define de (the) adition (addition) between A and B by the matriz (matrix) (A+B)(i,j)= (A(i,j)+B(i,j)), that is the elemnt (element) (i,j) from the matrix (A+B) , in other words, is the sum of the elements (i,j) of the matrix A and B.

Matrices have many properties, in order to name them we supose we have 3 matrices A,B,C with m rows and n columns, and we have k,r,s scalars that belongs to the scalar fiel (field) F we mention before. Some of those properties are the following: 1) A + B = B + A 2) ( (A + B) + C ) = ( A + ( B + C) ) 3) k ( A + B ) = k A + k B 4) (r + s) A = r A + s A 5) r(sA) = (rs) A

NOTE: I would place all the variables within quotation marks to recognize them with major facilit y