Let's consider the matrix
A = {{1, 1, 0}, {1, 0, 1}, {0, 1, 1}};
MatrixForm[A]
We can compute its eigenvalues and eigenvectors:
Eigenvalues[A]
Eigenvectors[A]
This yields the diagonalization of A:
T = Transpose[Eigenvectors[A]]; MatrixForm[T]
Inverse[T] . A . T // MatrixForm
% == DiagonalMatrix[Eigenvalues[A]]
We can solve linear systems:
LinearSolve[A, {1, 2, 3}]
A . %
In this case, the solution is unique:
NullSpace[A]
Let's consider a singular matrix:
B = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}};
MatrixRank[B]
s = LinearSolve[B, {1, 2, 3}]
NullSpace[B]
B . (RandomInteger[100] * %[[1]] + s)