Using LinearSolve instead of Inverse does not give a good enough precision

The problem is rounding error. The option Tolerance is one way Mathematica lets you deal with it in some functions.

SeedRandom[0];
n = 1000;
m = 10;
a = RandomReal[{-1, 1}, {n, m, m}];
b = RandomVariate[WishartMatrixDistribution[11, IdentityMatrix[10]], n];
result = Table[a[[i]].LinearSolve[b[[i]], Transpose[a[[i]]]], {i, 1, Length[a]}];

Total[PositiveDefiniteMatrixQ /@ result]
Total[SymmetricMatrixQ[#, Tolerance -> 1*^-10] & /@ result]
(*
  1000 True
  1000 True
*)

Here's why 1*^-10 is about the right tolerance:

ListPlot[#, PlotLabel -> Max[#]] &[
 $MachineEpsilon*Through[(LinearSolve /@ b)["ConditionNumber"]]
 ]

The condition number estimates the relative loss of precision due to LinearSolve. The maximum absolute value of the entries of the matrices b are near 1, so $MachineEpsilon is the right number to use to multiply the condition number. There is further rounding error in the matrix multiplication, so the tolerance needs to be a little greater than the estimated maximum error shown in the plot label above.

Using the CholeskyDecomposition explicitly not only seems to remove the problem, it is also faster: Moreover, this gets rid of one of the matrix-matrix multiplications and, probably more important, it requires onle one triangular matrix solve instead of two.

Notice also that I cast the matrix-matrix products in a way that Transpose[#].# & is applied last. This ensures that the resulting matrix is numerically symmetric and positive (semi-)definite.

n = 1000;
m = 50;
a = RandomReal[{-1, 1}, {n, m, m}];
b = RandomVariate[WishartMatrixDistribution[m + 1, IdentityMatrix[m]],
    n];

result = Table[
    a[[i]].LinearSolve[b[[i]], Transpose[a[[i]]]], {i, 1, 
     Length[a]}]; // AbsoluteTiming // First
result2 = Table[
    With[{L = Transpose[CholeskyDecomposition[b[[i]]]]},
     Transpose[#].# &[LinearSolve[L, Transpose[a[[i]]]]]
     ], {i, 1, Length[a]}]; // AbsoluteTiming // First

Total[SymmetricMatrixQ /@ result]
Total[SymmetricMatrixQ /@ result2]

0.437883

0.19313

529 False + 471 True

1000 True

Beware that CholeskyDecomposition does not apply pivoting which may also cause accuracy problems when small entries appear on the main diagonal.