(A = {{a, b}, {c, d}}) // MatrixForm
(v = {x, y}) // MatrixForm

Matricer ganget sammen

A . A

Vektor ganget matrice

v . A // MatrixForm

Determinant

Det[A]

Matrix powers (A . A)

MatrixPower[A, 2]

Transponeret (rækker til søjler og vice versa)

Transpose[A] // MatrixForm

Bestemmelse af egenværdier

(B = {{3, 0, 0}, {0, 3, 2}, {0, 2, 6}}) // 
      MatrixForm
Eigenvalues[B]

Egenvektorer

Eigenvectors[B]

Inverse[B] // MatrixForm

Karakteristisk polynomie

CharacteristicPolynomial[B, λ]

Løsning af lineært system

LinearSolve[B, {1, 0, 0}]

RowReduce[m]

If you have a square matrix m with a non-zero determinant, then you can always find a unique solution to the matrix equation m.x=b for any b. If, however, the matrix m has determinant zero, then there may be either no vector, or an infinite number of vectors x which satisfy m.x=b for a particular b. This occurs when the linear equations embodied in m are not independent.

When m has determinant zero, it is nevertheless always possible to find nonzero vectors x that satisfy m.x=0. The set of vectors x satisfying this equation form the null space or kernel of the matrix m. Any of these vectors can be expressed as a linear combination of a particular set of basis vectors, which can be obtained using NullSpace[m].

({{"Underdetermined"" ", "number of equations less than the number \
of variables; no solutions or many solutions may exist "}, {"Overdetermined"" ", "number of equations more than the number \
of variables; solutions may or may not exist "}, {"Nonsingular"" ", "number of independent equations equal to \
the number of variables, and determinant non\[Hyphen]zero; a unique solution \
exists"}, {"Consistent"" ", "at least one solution exists "}, {"Inconsistent"" ", "no solutions exist "}})

LP problem

NMaximize[{4 x - 
          3 y,  -x + 2 y ≤ 4, 2 x + 3 y ≤ 13, 
        x - y ≤ 4}, {x, y}]

Gram-schmidt

GramSchmidt[{{0, 1, -1/2 }, {1, 0, 0}, {0, 1, 2}}]  // 
      MatrixForm

M = {}

(A = {{1, 0, -1}, {0, 1, 1}, {-1, 1, 0}}) // 
      MatrixForm
CharacteristicPolynomial[A, λ]

Eigenvalues[A]

Eigenvectors[A]

A . Transpose[A]

Solve[a*(t + 1) + b*(t - 2) == 3*t + 9, {a, b}] // 
      FullSimplify

3/Sqrt[3] - 0.5*9/Sqrt[3] + 9/Sqrt[3] - 9

(∫_0^1)(18  t - 6) (18  t - 
            6) \[DifferentialD]t

(F = {{0, 0, 0, 1}, {0, 1, -1, 0}, {0, 0, 0, 0}, {1, 0, 0, 
              0}}) // MatrixForm

MatrixPower[F, 3] // MatrixForm

Eigenvalues[F]

Eigenvectors[F]