75032 Matrisemetoder 1996–06–06: Fasit

1a

L = [\begin{matrix}1 & 0 & 0 & 0 \\ s & 1 & 0 & 0 \\ 0 & s & 1 & 0 \\ 1 & −1 & 2 & 1 \end{matrix}] , U = [\begin{matrix}1 & −1 & 1 & −1 \\ 0 & 2 & 0 & 1 \\ 0 & 0 & s & 1 \\ 0 & 0 & 0 & s \end{matrix}] , rang A = \begin{cases}4 & s ≠ 0 \\ 3 & s = 0 \end{cases}

1b Alltid en løsning, unntatt når s = 0, t ≠ 1. 1c

rang [A b] = \begin{cases}4 & s ≠ 0 eller t ≠ 1 \\ 3 & s = 0, t = 1 \end{cases}

2 (Ikke entydig løsning)

Q_{1} = \frac{1}{\sqrt{3}} [\begin{matrix}1 & 0 & 1 & 1 \\ 1 & 1 & 0 & −1 \\ 1 & −1 & −1 & 0 \\ 0 & 1 & −1 & 1 \end{matrix}] , Σ = [\begin{matrix}\sqrt{3} & 0 \\ 0 & \sqrt{2} \\ 0 & 0 \\ 0 & 0 \end{matrix}] , Q_{2} = \frac{1}{\sqrt{2}} [\begin{matrix}1 & 1 \\ 1 & −1 \end{matrix}]

3a

Q = \frac{1}{2} [\begin{matrix}1 & 1 & 1 \\ 1 & −1 & −1 \\ 1 & −1 & 1 \\ 1 & 1 & −1 \end{matrix}] , R = [\begin{matrix}2 & −2 & 0 \\ 0 & 4 & −2 \\ 0 & 0 & 2 \end{matrix}]

3b

[ /_{4}^{9} ; −^{1}/_{4} ; −^{1}/_{2}]

3c

\frac{1}{2} [\begin{matrix}1 \\ 1 \\ −1 \\ −1 \end{matrix}]

4

∑_{m=1}^{n} b_{m} ∏_{k=1}^{m−1} a_{k}