ZeroTest

is an option to various linear algebra functions that gives a function to use in testing whether symbolic expressions should be treated as zero.

Details

ZeroTest->Automatic effectively tests for zeros using a version of PossibleZeroQ.

The automatic ZeroTest used by LinearSolve does not detect that mat is non-singular:

Wolfram Language code:

a = RootApproximant[N[Pi, 10000], 20];
mat = {{1, 0}, {1, a - Pi}};
LinearSolve[mat, {1, 2}]

Wolfram Language code: N[N[a - Pi, 20000], 20]//Quiet

With a stronger zero test, LinearSolve gives the correct answer:

Wolfram Language code: LinearSolve[mat, {1, 2}, ZeroTest -> (N[#, 20000] == 0&)]

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