P = [ − 1 4 5 0 2 − 3 0 0 8 ] P=\begin{bmatrix}
-1&4&5
\\
0&2&-3
\\
0&0&8
\end{bmatrix} P = ⎣ ⎡ − 1 0 0 4 2 0 5 − 3 8 ⎦ ⎤
p ( λ ) = det ( P − λ I ) = ∣ − 1 − λ 4 5 0 2 − λ − 3 0 0 8 − λ ∣ = ( − 1 − λ ) ( 2 − λ ) ( 8 − λ ) 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 ) = det ( P − λ I ) = ∣ ∣ − 1 − λ 0 0 4 2 − λ 0 5 − 3 8 − λ ∣ ∣ = ( − 1 − λ ) ( 2 − λ ) ( 8 − λ )
p ( λ ) = − x 3 + 9 x 2 − 6 x − 16 p(\lambda )=-x^3+9x^2-6x-16 p ( λ ) = − x 3 + 9 x 2 − 6 x − 16
− P 3 + 9 P 2 − 6 P − 16 I = 0 -P^3+9P^2-6P-16I=0 − P 3 + 9 P 2 − 6 P − 16 I = 0
P ( − P 2 + 9 P − 6 I ) = 16 I P(-P^2+9P-6I)=16I P ( − P 2 + 9 P − 6 I ) = 16 I
16 P − 1 = − P 2 + 9 P − 6 I = − [ 1 4 23 0 4 − 30 0 0 64 ] + 9 [ − 1 4 5 0 2 − 3 0 0 8 ] − [ 6 0 0 0 6 0 0 0 6 ] 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} 16 P − 1 = − P 2 + 9 P − 6 I = − ⎣ ⎡ 1 0 0 4 4 0 23 − 30 64 ⎦ ⎤ + 9 ⎣ ⎡ − 1 0 0 4 2 0 5 − 3 8 ⎦ ⎤ − ⎣ ⎡ 6 0 0 0 6 0 0 0 6 ⎦ ⎤
16 P − 1 = [ − 16 32 22 0 8 3 0 0 2 ] 16P^{-1}=\begin{bmatrix}
-16&32&22
\\
0&8&3
\\
0&0&2
\end{bmatrix} 16 P − 1 = ⎣ ⎡ − 16 0 0 32 8 0 22 3 2 ⎦ ⎤
P − 1 = 1 16 [ − 16 32 22 0 8 3 0 0 2 ] P^{-1}=\frac{1}{16}\begin{bmatrix}
-16&32&22
\\
0&8&3
\\
0&0&2
\end{bmatrix} P − 1 = 16 1 ⎣ ⎡ − 16 0 0 32 8 0 22 3 2 ⎦ ⎤
Let [ x 1 x 2 x 3 ] = a ⋅ [ − 1 0 0 ] + b ⋅ [ 4 2 0 ] + c ⋅ [ 5 − 3 8 ] = P [ a b c ] \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} ⎣ ⎡ x 1 x 2 x 3 ⎦ ⎤ = a ⋅ ⎣ ⎡ − 1 0 0 ⎦ ⎤ + b ⋅ ⎣ ⎡ 4 2 0 ⎦ ⎤ + c ⋅ ⎣ ⎡ 5 − 3 8 ⎦ ⎤ = P ⎣ ⎡ a b c ⎦ ⎤ .
Then [ a b c ] = P − 1 [ x 1 x 2 x 3 ] = [ − x 1 + 2 x 2 + 11 8 x 3 1 2 x 2 + 3 16 x 3 1 8 x 3 ] \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} ⎣ ⎡ a b c ⎦ ⎤ = P − 1 ⎣ ⎡ x 1 x 2 x 3 ⎦ ⎤ = ⎣ ⎡ − x 1 + 2 x 2 + 8 11 x 3 2 1 x 2 + 16 3 x 3 8 1 x 3 ⎦ ⎤ .
So, [ x 1 x 2 x 3 ] = ( − x 1 + 2 x 2 + 11 8 x 3 ) ⋅ [ − 1 0 0 ] + ( 1 2 x 2 + 3 16 x 3 ) ⋅ [ 4 2 0 ] + 1 8 x 3 ⋅ [ 5 − 3 8 ] \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} ⎣ ⎡ x 1 x 2 x 3 ⎦ ⎤ = ( − x 1 + 2 x 2 + 8 11 x 3 ) ⋅ ⎣ ⎡ − 1 0 0 ⎦ ⎤ + ( 2 1 x 2 + 16 3 x 3 ) ⋅ ⎣ ⎡ 4 2 0 ⎦ ⎤ + 8 1 x 3 ⋅ ⎣ ⎡ 5 − 3 8 ⎦ ⎤ .
Answers:
P − 1 = 1 16 [ − 16 32 22 0 8 3 0 0 2 ] P^{-1}=\frac{1}{16}\begin{bmatrix}
-16&32&22
\\
0&8&3
\\
0&0&2
\end{bmatrix} P − 1 = 16 1 ⎣ ⎡ − 16 0 0 32 8 0 22 3 2 ⎦ ⎤
[ x 1 x 2 x 3 ] = ( − x 1 + 2 x 2 + 11 8 x 3 ) ⋅ [ − 1 0 0 ] + ( 1 2 x 2 + 3 16 x 3 ) ⋅ [ 4 2 0 ] + 1 8 x 3 ⋅ [ 5 − 3 8 ] \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} ⎣ ⎡ x 1 x 2 x 3 ⎦ ⎤ = ( − x 1 + 2 x 2 + 8 11 x 3 ) ⋅ ⎣ ⎡ − 1 0 0 ⎦ ⎤ + ( 2 1 x 2 + 16 3 x 3 ) ⋅ ⎣ ⎡ 4 2 0 ⎦ ⎤ + 8 1 x 3 ⋅ ⎣ ⎡ 5 − 3 8 ⎦ ⎤
Comments