Lists and Matrices

Question

I have a matrix of the form:

 matrix ={{a,b},{10*a,10*b}}.

I want to evaluate this matrix for a set of values. For example:

 list = {{1,2},{3,4},{5,6},{7,8},{9,10}},

where each sublist has the values of $a$ and $b$ respectively. How would I go about evaluating the matrix at the values in the list?

---

Edit:

More specifically, I have a list of the form:

OptimalPoints = {
{
 2.2575422669022256316146199310035044964832661776461`15.*^-8, 
 1.57079639622776237973336943808513436304`15.
},
{
 6.5030057578379460133211810893852417847654553996698`15.*^-8, 
 1.57079630003754406967047630250839359743`15.
},
{
 0, 
 1.57079625737764823615943901490196281871`15.
}
}

and the matrix is derived from the code:

J = 1;

Clear[ψ]
ψ[α_, χ_] := Exp[I α]*Tan[χ/2];



newcoherentstate[α_, χ_, m_] := ((ψ[α, χ])/(1 + ψ[α, χ]*
         Simplify[
          Conjugate[ψ[α, χ]], 
          Assumptions -> {Element[α, Reals], Element[χ, Reals]}
         ]))^(J) *(ψ[α, χ])^(m)*Sqrt[Binomial[2 J, J + m]];

TempNewMatrix [α_, χ_] := Simplify[
          Table[
               newcoherentstate[α, χ, m], 
               {m, J, -J, -1}
              ], 
          Assumptions -> {Element[α, Reals], Element[χ, Reals]}
        ];

NewMatrix [α_, χ_] := ArrayReshape[TempNewMatrix [α, χ], {2*J + 1, 1}];

NewMatrix1[α_, χ_] := ArrayReshape[NewMatrix[α, χ], {1, 2*J + 1}]

NewMatrixC[α_, χ_] := Simplify[
   Conjugate[NewMatrix1[α, χ]],
   Assumptions ->{Element[α, Reals], Element[χ, Reals]} 
];

NewProjector[α_, χ_] = Flatten[{NewMatrix[α, χ] .NewMatrixC[α, χ]}, 1];

When I run,

{{#1, #2}, NewProjector[α, χ]} & @@@ 
  OptimalPoints // MatrixForm

I don't get the desired list of matrices evaluated at points in the list optimal points

Answer 1

A clean way to do this is to define a pure function and then Apply it to your list:

{{#1, #2}, {10 #1, 10 #2}} & @@@ list

{{{1, 2}, {10, 20}}, {{3, 4}, {30, 40}}, {{5, 6}, {50, 60}}, {{7, 8}, {70, 80}}, {{9, 10}, {90, 100}}}

Alternatively, you can create your own Function and Apply it to your list.

matrixFcn[a_, b_] := {{a, b}, {10 a, 10 b}}
matrixFcn @@@ list

The same output is generated.