Jen

Laregst entry in matrix.

m=12;
A= RandomReal[{-1,1}, {m,m}];
A=A+Transpose[A]
TestMat=LowerTriangularize[Abs[A],-1]
MaxVal=Max[TestMat]
Position[TestMat,MaxVal]

Largest entry in a specified row!

m=12;
i=6;
A= RandomReal[{-1,1}, {m,m}]
OurRow=Abs[A[[i]]]
MaxVal=Max[OurRow]
Position[OurRow,MaxVal]

Largest k entries in a specified row first.

m=12;
i=6;
k=3;
A= RandomReal[{-1,1}, {m,m}]
OurRow=Abs[A[[i]]]
OurBigs=Ordering[OurRow,-k]
OurRow[[OurBigs]]

Largest k entries in a matrix

m=12;
A= RandomReal[{-1,1}, {m,m}];
A=A+Transpose[A]
TestMat=LowerTriangularize[Abs[A],-1];
k=4;
Thing=Ordering[TestMat,k, Max[#1]>Max[#2]&]

Clear[FindMaxInVec]
FindMaxInVec[x_]:= Position[x,Max[x]][[1,1]]
FindMaxInVec[{1.1614968234777807`,0.07536543558863018`,0.2951807095580832`,0.821704015037203`,1.5770034045514514`,0.2752890292528325`,1.1597790647062998`,0.24853417990316107`,0.`,0.`,0.`,0.`}]

Map[FindMaxInVec,TestMat[[Thing]]]

Ordering[TestMat,k, Max[#1]>Max[#2]&]

?Ordering

TestMat[[2]]
TestMat[[6]]
TestMat[[5]]