Matrix operation (simplify)

Question

I had a $6\times 6$ matrix

{{1., 0., 0., 0., 0, 0},
 {0, 0.538752, 0.382352, 0.0788956, 2.35142*10^-7, 0}, 
 {0, 0.121876, 0.449018, 0.405176, 0.0239311, 0},
 {0, 0.0147839, 0.264751, 0.596752, 0.123713, 0},
 {0, 0, 0.0294091, 0.381909, 0.563762, 0.0249205},
 {0, 0, 0., 0., 0., 1.}}

and I wanted to multiply row 3 by 0.8, row 4 by 1.2 and divide each column by the sum of each column except the first and last column.

Instead of doing it manually by editing it on the matrix, is there an alternative or Mathematica code to do so?

Answer 1

Operation on m itself:

m[[{3, 4}]] *= {.8, 1.2};

m[[All, 2 ;; -2]] = Transpose[Normalize[#, Total] & /@ Transpose[m[[All, 2 ;; -2]]]];

m // MatrixForm

after a talk, if you want to normalize (with Total norm) rows then skip transposing:

m[[2 ;; -2]] = Normalize[#, Total] & /@ m[[2 ;; -2]];

and if you don't want to modify m you can do:

Fold[
 MapAt[#2[[1]], #, #2[[2]]] &,
 m,
 {{.8 # &, 3},
  {1.2 # &, 4},
  {Normalize[#, Total] &, 2 ;; -2}
  }
 ]

Answer 2

list = {{1., 0., 0., 0., 0, 0}, {0, 0.538752, 0.382352, 0.0788956, 
    2.35142*10^-7, 0}, {0, 0.121876, 0.449018, 0.405176, 0.0239311, 
    0}, {0, 0.0147839, 0.264751, 0.596752, 0.123713, 0}, {0, 0, 
    0.0294091, 0.381909, 0.563762, 0.0249205}, {0, 0, 0., 0., 0., 
    1.}};  
list2 = {1, 1, 0.8, 1.2, 1, 1} list;
Transpose[MapAt[#/Total[#] &, Transpose[list2], {{2}, {3}, {4}, {5}}]]
(*{{1., 0., 0., 0., 0., 0}, {0, 0.823788, 0.351208, 0.0525604, 
  3.21512*10^-7, 0}, {0., 0.149085, 0.329955, 0.215943, 0.026177, 
  0.}, {0., 0.0271267, 0.291823, 0.477068, 0.202985, 0.}, {0, 0., 
  0.0270136, 0.254428, 0.770838, 0.0249205}, {0, 0., 0., 0., 0., 1.}}*)