Question #274145

Let P = [ -1 4 5] . Determine P^-1 using


[ 0 2 -3]


[ 0 0 8]


Cayley- Hamilton theorem. Further use P^-1 to express (x1, x2, x3) in terms of (-1,0,0), (4,2,0), ( 5,-3,8)

1
Expert's answer
2021-12-02T12:17:02-0500

P=[145023008]P=\begin{bmatrix} -1&4&5 \\ 0&2&-3 \\ 0&0&8 \end{bmatrix}

p(λ)=det(PλI)=1λ4502λ3008λ=(1λ)(2λ)(8λ)p(\lambda)=\det (P-\lambda I)=\begin{vmatrix} -1-\lambda&4&5 \\ 0&2-\lambda &-3 \\ 0&0&8-\lambda \end{vmatrix} =(-1-\lambda )(2-\lambda )(8-\lambda)

p(λ)=x3+9x26x16p(\lambda )=-x^3+9x^2-6x-16


P3+9P26P16I=0-P^3+9P^2-6P-16I=0

P(P2+9P6I)=16IP(-P^2+9P-6I)=16I

16P1=P2+9P6I=[142304300064]+9[145023008][600060006]16P^{-1}=-P^2+9P-6I=-\begin{bmatrix} 1&4&23 \\ 0&4&-30 \\ 0&0&64 \end{bmatrix}+9\begin{bmatrix} -1&4&5 \\ 0&2&-3 \\ 0&0&8 \end{bmatrix}-\begin{bmatrix} 6&0&0 \\ 0&6&0 \\ 0&0&6 \end{bmatrix}


16P1=[163222083002]16P^{-1}=\begin{bmatrix} -16&32&22 \\ 0&8&3 \\ 0&0&2 \end{bmatrix}


P1=116[163222083002]P^{-1}=\frac{1}{16}\begin{bmatrix} -16&32&22 \\ 0&8&3 \\ 0&0&2 \end{bmatrix}


Let [x1x2x3]=a[100]+b[420]+c[538]=P[abc]\begin{bmatrix} x_1 \\ x_2\\ x_3 \end{bmatrix}=a\cdot \begin{bmatrix}-1\\0\\0 \end{bmatrix}+ b\cdot \begin{bmatrix}4\\2\\0 \end{bmatrix}+ c\cdot \begin{bmatrix}5\\-3\\8 \end{bmatrix}=P\begin{bmatrix}a\\b\\c\end{bmatrix} .


Then [abc]=P1[x1x2x3]=[x1+2x2+118x312x2+316x318x3]\begin{bmatrix} a\\b\\c \end{bmatrix}=P^{-1}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}= \begin{bmatrix} -x_1+2x_2+\tfrac{11}{8} x_3 \\ \tfrac{1}{2}x_2+\tfrac{3}{16}x_3 \\ \tfrac{1}{8}x_3 \end{bmatrix} .


So, [x1x2x3]=(x1+2x2+118x3)[100]+(12x2+316x3)[420]+18x3[538]\begin{bmatrix} x_1 \\ x_2\\ x_3 \end{bmatrix}=(-x_1+2x_2+\tfrac{11}{8}x_3)\cdot \begin{bmatrix}-1\\0\\0 \end{bmatrix}+ (\tfrac{1}{2}x_2+\tfrac{3}{16}x_3)\cdot \begin{bmatrix}4\\2\\0 \end{bmatrix}+ \tfrac{1}{8}x_3\cdot \begin{bmatrix}5\\-3\\8 \end{bmatrix} .




Answers:

P1=116[163222083002]P^{-1}=\frac{1}{16}\begin{bmatrix} -16&32&22 \\ 0&8&3 \\ 0&0&2 \end{bmatrix}


[x1x2x3]=(x1+2x2+118x3)[100]+(12x2+316x3)[420]+18x3[538]\begin{bmatrix} x_1 \\ x_2\\ x_3 \end{bmatrix}=(-x_1+2x_2+\tfrac{11}{8}x_3)\cdot \begin{bmatrix}-1\\0\\0 \end{bmatrix}+ (\tfrac{1}{2}x_2+\tfrac{3}{16}x_3)\cdot \begin{bmatrix}4\\2\\0 \end{bmatrix}+ \tfrac{1}{8}x_3\cdot \begin{bmatrix}5\\-3\\8 \end{bmatrix}


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