How can I improve displaying repetitive equations

I’m looking to improve my report quality by making equations more clear. As the equations I’m stating are repetitive a couple of times throughout the report I thought it would be better to use matrix notation.

Currently the equations look like this:

begin{align}
    dCA_{p,1} =  - &1 cdot A - 2 cdot B + 3 cdot C nonumber
                 + &4 cdot A cdot B + 5 cdot A^2 - 6 cdot C ^2label{eq:dCA_p_1_2}
    dCA_{m,1} =  &8 cdot A - 9 cdot B + 5 cdot C nonumber
                - &5 cdot A cdot B - 4 cdot A^2 + 1 cdot B^2label{eq:dCA_m_1_2}
                - &3 cdot C ^2nonumber
    dCA_{l,1} =  &3 - 3 cdot A - 2 cdot B - 3 cdot C  + 3 cdot Dnonumber
                + &5 cdot Acdot B - 3 cdot Acdot D label{eq:dCA_l_1_2}
                - &1 cdot Bcdot D - 2 cdot C cdot D + 3 cdot B^2nonumber
                + &2 cdot C ^2 + 6 cdot D^2nonumber
end{align}

---

One can clearly notice the repetitive parameters A , B , C and D. Therefore I tried the following approach:

begin{equation}
  left( begin{array}{c}
    dCA_{p,1}
    dCA_{m,1}
    dCA_{l,1}  end{array}right)
  =
  left(begin{array}{cccccccccccccc}
    1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 
    1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 
    1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 
  end{array}right)   
  left(begin{array}{c}
      A
      B
      C
      D
      Acdot B
      Acdot C
      Acdot D
      Bcdot C
      Bcdot D
      Ccdot D
      A^2
      B^2
      C^2
      D^2 end{array}right)
end{equation}

This output is however very wide and long. Already with single rounded numbers. Actual numbers would be a 4 digit number, for example: 2.314.

---

Is the last approach the way to go or are there other possibilities? Maybe there is a way to reduce the length by rewriting the squared signs in another way. I’m not sure how accomplish this.

Replacing the array with pmatrix gives a slightly slimmer expression, yet still very wide…

begin{equation}
  left( begin{array}{c}
    dCA_{p,1}
    dCA_{m,1}
    dCA_{l,1}  end{array}right)
  =
  begin{pmatrix}
    1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 
    1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 
    1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 
  end{pmatrix}   left(begin{array}{c}
      A
      B
      C
      D
      Acdot B
      Acdot C
      Acdot D
      Bcdot C
      Bcdot D
      Ccdot D
      A^2
      B^2
      C^2
      D^2 end{array}right)
end{equation}

---

As proposed by Mico, as far as i understood:

begin{equation}
  begin{pmatrix}
    dCA_{p,1}
    dCA_{m,1}
    dCA_{l,1}  
  pmatrix{pmatrix}
  =
  begin{pmatrix}
    1 & 2 & 3 & 4  
    1 & 2 & 3 & 4  
    1 & 2 & 3 & 4  
  end{pmatrix}   
  begin{pmatrix}
      A
      B
      C
      D
   end{pmatrix}
   +   
   begin{pmatrix}
        1 & 2 & 3 & 4 & 5 & 6 
        1 & 2 & 3 & 4 & 5 & 6 
        1 & 2 & 3 & 4 & 5 & 6  
   end{pmatrix}   
   begin{pmatrix}
   Acdot B
   Acdot C
   Acdot D
   Bcdot C
   Bcdot D
   Ccdot D
   begin{pmatrix} 
   +  
   begin{pmatrix}
    1 & 2 & 3 & 4  
    1 & 2 & 3 & 4  
    1 & 2 & 3 & 4  
   end{pmatrix}   
   begin{pmatrix}
   A^2
   B^2
   C^2   
   D^2 
   begin{pmatrix}
end{equation}

documentclass{article}
usepackage{mathtools} % for 'pmatrix*' env.
setcounter{MaxMatrixCols}{14}
newcommandvn[1]{textit{#1}}
begin{document}

begin{equation}
begin{split}
  begin{pmatrix}
    dvn{CA}_{p,1}
    dvn{CA}_{m,1}
    dvn{CA}_{l,1}  
  end{pmatrix}
  &=
  begin{pmatrix*}[r]
   -1 & -2 &  3 & 0 
    8 & -9 &  5 & 0 
   -3 & -2 & -3 & 3  
  end{pmatrix*}  
  begin{pmatrix}
    A  B  C  D
  end{pmatrix} 
  &quad+
  begin{pmatrix*}[r]
    4 & 0 &  0 & 0 &  0 &  0 
   -5 & 0 &  0 & 0 &  0 &  0 
    0 & 0 &  0 & 0 & -1 & -2 
  end{pmatrix*}  
  begin{pmatrix}
    A B  A C  A D  B C  B D  C D
  end{pmatrix} 
  &quad+
  begin{pmatrix*}[r]
   5 & 0 & -6 &  0 
   0 & 0 & -3 &  0 
   0 & 0 &  2 &  6 
  end{pmatrix*}    
  begin{pmatrix}
      A^2  B^2  C^2  D^2 
  end{pmatrix}
end{split}
end{equation}
end{document}

documentclass{article}
usepackage{amsmath}

setcounter{MaxMatrixCols}{15}

begin{document}

begin{align}
dCA_{p,1} &= - A - 2 B + 3 C notag 
          &qquad + 4 A B notag 
          &qquad + 5 A^2 - 6 C ^2 label{eq:dCA_p_1_2}
[1ex]
dCA_{m,1} &= 8 A - 9 B + 5 C notag 
          &qquad - 5 A B notag 
          &qquad - 4 A^2 + 1 B^2 - 3 C ^2 label{eq:dCA_m_1_2}
[1ex]
dCA_{l,1} &=  3 - 3 A - 2 B - 3 C  + 3 D notag 
          &qquad + 5 AB - 3 AD - BD - 2 C D notag 
          &qquad + 3 B^2 + 2 C^2 + 6 D^2 label{eq:dCA_l_1_2}
end{align}

begin{equation}
  begin{pmatrix}
  dCA_{p,1}
  dCA_{m,1}
  dCA_{l,1}
  end{pmatrix}
  = MT
end{equation}
where
begin{align*}
M &= begin{pmatrix}
     % 0    A    B    C    D   AB   AC   AD   BC   BD   CD   A2   B2   C2   D2
       0 & -1 & -2 &  3 &  0 &  4 &  0 &  0 &  0 &  0 &  0 &  5 &  0 &  0 &  0 
       0 &  8 & -9 &  5 &  0 & -5 &  0 &  0 &  0 &  0 &  0 & -4 &  1 & -3 &  0 
       3 & -3 & -2 & -3 &  3 &  5 &  0 & -3 &  0 & -1 & -2 &  0 &  3 &  2 &  6
  end{pmatrix}

T &=
     addtolength{arraycolsep}{-2pt}
     begin{pmatrix}
     1 & A & B & C & D & AB & AC & AD & BC & BD & CD & A^2 & B^2 & C^2 & D^2
     end{pmatrix}^T
end{align*}
are the necessary matrices.

end{document}