Let’s consider the matrix
>> A = {{1, 1, 0}, {1, 0, 1}, {0, 1, 1}};
The derivatives are
>> 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)