Finding an idiomatic way to express a system of linear equations

Question

I have an $n$ x $n$ matrix $T$, where $t_{ij}$ is the element at the $i\,$th row and $j\,$th column, and unknowns $x_i$, $1\leq i \leq n$, where $$x_{n} = 1$$ $$x_{n-1} = 0$$ $$x_i = \sum_{1\leq j \leq n}{t_{ij}x_j};\;1\leq i \leq n-2$$ and am trying to package up this problem statement for use in LinearSolve. I can get everything working satisfactorily with

LinearSolve[T - DiagonalMatrix[Join[Table[1, {n - 2}], {0, 0}]], Append[Table[0, {n - 1}], 1]]

but feel like I've made things more complicated than they need to be.

Is there a better, more compact — perhaps more idiomatic — way to express this problem?

Answer 1

This is but a mild variant. I'll create an example using n=4.

SeedRandom[1111];
n = 4;
tmat = RandomInteger[{-10, 10}, {n - 2, n}]

(* Out[190]= {{-8, 5, -3, -8}, {-4, 2, 4, 2}} *)

LinearSolve[IdentityMatrix[n] - 
  Join[tmat, {ConstantArray[0, n], ConstantArray[0, n]}], UnitVector[n, n]]

(* Out[196]= {18/11, 50/11, 0, 1} *)

An alternative might be to use NullSpace but then you will need to renormalize based on the last value.

NullSpace[IdentityMatrix[n] - Join[tmat, {ConstantArray[0, n], UnitVector[n, n]}]]

(* Out[193]= {{18, 50, 0, 11}} *)

Answer 2

A bit shorter and more expressive:

LinearSolve[T - PadRight[IdentityMatrix[n - 2], {n, n}], PadLeft[{1}, n]]