TU Wien:Mathematik 1 UE (diverse)/Übungen WS06/Beispiel 555

Berechnen Sie zur folgenden Matrix A mit Eintragungen aus $\mathbb {R}$ die Matrix $A^{3}$.

$A={\begin{pmatrix}0&0&0\\a&0&0\\b&c&0\end{pmatrix}}$

Lösungsvorschlag von mnemetz

$A^{3}=\underbrace {{\begin{pmatrix}0&0&0\\a&0&0\\b&c&0\end{pmatrix}}*{\begin{pmatrix}0&0&0\\a&0&0\\b&c&0\end{pmatrix}}} _{\begin{matrix}&&&0&0&0\\&&&a&0&0\\&&&b&c&0\\0&0&0&{\mathit {0}}&{\mathit {0}}&{\mathit {0}}\\a&0&0&{\mathit {0}}&{\mathit {0}}&{\mathit {0}}\\b&c&0&{\mathit {a*c}}&{\mathit {0}}&{\mathit {0}}\\\end{matrix}}*{\begin{pmatrix}0&0&0\\a&0&0\\b&c&0\end{pmatrix}}=$

$\underbrace {{\begin{pmatrix}0&0&0\\0&0&0\\a*c&0&0\end{pmatrix}}*{\begin{pmatrix}0&0&0\\a&0&0\\b&c&0\end{pmatrix}}} _{\begin{matrix}&&&0&0&0\\&&&a&0&0\\&&&b&c&0\\0&0&0&{\mathit {0}}&{\mathit {0}}&{\mathit {0}}\\0&0&0&{\mathit {0}}&{\mathit {0}}&{\mathit {0}}\\a*c&0&0&{\mathit {0}}&{\mathit {0}}&{\mathit {0}}\\\end{matrix}}=$

$={\begin{pmatrix}{\mathit {0}}&{\mathit {0}}&{\mathit {0}}\\{\mathit {0}}&{\mathit {0}}&{\mathit {0}}\\{\mathit {0}}&{\mathit {0}}&{\mathit {0}}\\\end{pmatrix}}$

Verifizierung mit MATLAB

Symbolische Variablen a,b,c setzen

  >> a = sym('a');
  >> b = sym('b');
  >> c = sym('c');

Matrix definieren

  >> A = [0 0 0; a 0 0; b c 0]
     A =
     [ 0, 0, 0]
     [ a, 0, 0]
     [ b, c, 0]

A*A

  >> A*A
    ans =
     [   0,   0,   0]
     [   0,   0,   0]
     [ c*a,   0,   0]

A*A*A

  >> A*A*A
    ans = 
     [ 0, 0, 0]
     [ 0, 0, 0]
     [ 0, 0, 0]