Answer on Question #82533 – Math – Linear Algebra
Question
a. Find a basis for the span of the following set of vectors
|1 | |-4 | |1 | |5 | |2 | |-1 | |11|
|3 | |-12| |-4| |8| |11| |-7| |52|
|-1| |4| -5| -17| -21| 19| -80|
|2| -8| 4| 16| 17| -3| 82|
Solution
a
( 1 − 4 1 5 2 − 1 11 3 − 12 − 4 8 11 − 7 52 − 1 4 − 5 − 17 − 21 19 − 80 2 − 8 4 16 17 − 3 82 ) → r 2 = r 2 + ( − 3 ) r 1 r 3 = r 3 + r 1 r 4 = r 4 + ( − 2 ) r 1 \left(\begin{array}{cccccc}
1 & -4 & 1 & 5 & 2 & -1 & 11 \\
3 & -12 & -4 & 8 & 11 & -7 & 52 \\
-1 & 4 & -5 & -17 & -21 & 19 & -80 \\
2 & -8 & 4 & 16 & 17 & -3 & 82
\end{array}\right)
\rightarrow
\begin{array}{c}
r2 = r2 + (-3)r1 \\
r3 = r3 + r1 \\
r4 = r4 + (-2)r1
\end{array} ⎝ ⎛ 1 3 − 1 2 − 4 − 12 4 − 8 1 − 4 − 5 4 5 8 − 17 16 2 11 − 21 17 − 1 − 7 19 − 3 11 52 − 80 82 ⎠ ⎞ → r 2 = r 2 + ( − 3 ) r 1 r 3 = r 3 + r 1 r 4 = r 4 + ( − 2 ) r 1 → ( 1 − 4 1 5 2 − 1 11 0 0 − 7 − 7 5 − 4 19 0 0 − 4 − 12 − 19 18 − 69 0 0 2 6 13 − 1 60 ) → r 1 = r 1 + ( 1 7 ) r 2 r 3 = r 3 + ( − 4 7 ) r 2 r 4 = r 4 + ( 2 7 ) r 2 \rightarrow \left(\begin{array}{cccccc}
1 & -4 & 1 & 5 & 2 & -1 & 11 \\
0 & 0 & -7 & -7 & 5 & -4 & 19 \\
0 & 0 & -4 & -12 & -19 & 18 & -69 \\
0 & 0 & 2 & 6 & 13 & -1 & 60
\end{array}\right)
\rightarrow
\begin{array}{c}
r1 = r1 + \left(\frac{1}{7}\right) r2 \\
r3 = r3 + \left(\frac{-4}{7}\right) r2 \\
r4 = r4 + \left(\frac{2}{7}\right) r2
\end{array} → ⎝ ⎛ 1 0 0 0 − 4 0 0 0 1 − 7 − 4 2 5 − 7 − 12 6 2 5 − 19 13 − 1 − 4 18 − 1 11 19 − 69 60 ⎠ ⎞ → r 1 = r 1 + ( 7 1 ) r 2 r 3 = r 3 + ( 7 − 4 ) r 2 r 4 = r 4 + ( 7 2 ) r 2 → ( 19 7 − 11 7 96 7 1 − 4 0 4 5 − 4 19 0 0 − 7 − 7 153 7 142 7 − 559 7 0 0 0 4 101 7 − 15 7 458 7 ) ← r 1 = r 1 + ( 1 2 ) r 3 r 2 = r 2 + ( − 7 8 ) r 3 r 4 = r 4 + ( 1 2 ) r 3 \rightarrow \left(\begin{array}{cccccc}
& & & & \frac{19}{7} & -\frac{11}{7} & \frac{96}{7} \\
1 & -4 & 0 & 4 & 5 & -4 & 19 \\
0 & 0 & -7 & -7 & \frac{153}{7} & \frac{142}{7} & -\frac{559}{7} \\
0 & 0 & 0 & 4 & \frac{101}{7} & -\frac{15}{7} & \frac{458}{7}
\end{array}\right)
\leftarrow
\begin{array}{c}
r1 = r1 + \left(\frac{1}{2}\right) r3 \\
r2 = r2 + \left(\frac{-7}{8}\right) r3 \\
r4 = r4 + \left(\frac{1}{2}\right) r3
\end{array} → ⎝ ⎛ 1 0 0 − 4 0 0 0 − 7 0 4 − 7 4 7 19 5 7 153 7 101 − 7 11 − 4 7 142 − 7 15 7 96 19 − 7 559 7 458 ⎠ ⎞ ← r 1 = r 1 + ( 2 1 ) r 3 r 2 = r 2 + ( 8 − 7 ) r 3 r 4 = r 4 + ( 2 1 ) r 3 \rightarrow \left(\begin{array}{cccccc}
& & & & -\frac{115}{14} & \frac{60}{7} & -\frac{367}{14} \\
1 & -4 & 0 & 0 & \frac{193}{8} & -\frac{87}{4} & \frac{711}{8} \\
0 & 0 & -7 & 0 & \frac{193}{8} & -\frac{87}{4} & \frac{711}{8} \\
0 & 0 & 0 & -8 & -\frac{153}{7} & \frac{142}{7} & -\frac{559}{7} \\
0 & 0 & 0 & 0 & \frac{7}{2} & \frac{8}{2}
\end{array}\right)
\leftarrow
\begin{array}{c}
r1 = r1 + \left(\frac{115}{49}\right) r4 \\
r2 = r2 + \left(\frac{-193}{28}\right) r4 \\
r3 = r3 + \left(\frac{306}{49}\right) r4
\end{array}
\right) → ( 1340 49 1648 49 1 − 4 0 0 0 2153 28 − 2433 28 0 0 − 7 0 0 2153 28 3890 49 0 0 0 0 2 2 3442 49 51 2 ) \rightarrow \left(\begin{array}{cccccc}
& & & & \frac{1340}{49} & \frac{1648}{49} \\
1 & -4 & 0 & 0 & 0 & \frac{2153}{28} & -\frac{2433}{28} \\
0 & 0 & -7 & 0 & 0 & \frac{2153}{28} & \frac{3890}{49} \\
0 & 0 & 0 & 0 & \frac{2}{2} & \frac{3442}{49} & \frac{51}{2}
\end{array}\right) → ⎝ ⎛ 1 0 0 − 4 0 0 0 − 7 0 0 0 0 49 1340 0 0 2 2 49 1648 28 2153 28 2153 49 3442 − 28 2433 49 3890 2 51 ⎠ ⎞
The first, the third, the fourth, the fifth columns are linearly independent, hence a basis is:
B = { ( 1 3 − 1 2 ) , ( 1 − 4 − 5 4 ) , ( 5 8 − 17 16 ) , ( 2 11 − 21 17 ) } B = \left\{\left(\begin{array}{c}
1 \\
3 \\
-1 \\
2
\end{array}\right), \left(\begin{array}{c}
1 \\
-4 \\
-5 \\
4
\end{array}\right), \left(\begin{array}{c}
5 \\
8 \\
-17 \\
16
\end{array}\right), \left(\begin{array}{c}
2 \\
11 \\
-21 \\
17
\end{array}\right)\right\} B = ⎩ ⎨ ⎧ ⎝ ⎛ 1 3 − 1 2 ⎠ ⎞ , ⎝ ⎛ 1 − 4 − 5 4 ⎠ ⎞ , ⎝ ⎛ 5 8 − 17 16 ⎠ ⎞ , ⎝ ⎛ 2 11 − 21 17 ⎠ ⎞ ⎭ ⎬ ⎫
Answer a.: basis { ( 1 3 − 1 2 ) , ( 1 − 4 − 5 4 ) , ( 5 8 − 17 16 ) , ( 2 11 − 21 17 ) } . \left\{\left( \begin{array}{l}1\\ 3\\ -1\\ 2 \end{array} \right),\left( \begin{array}{l}1\\ -4\\ -5\\ 4 \end{array} \right),\left( \begin{array}{l}5\\ 8\\ -17\\ 16 \end{array} \right),\left( \begin{array}{l}2\\ 11\\ -21\\ 17 \end{array} \right)\right\}. ⎩ ⎨ ⎧ ⎝ ⎛ 1 3 − 1 2 ⎠ ⎞ , ⎝ ⎛ 1 − 4 − 5 4 ⎠ ⎞ , ⎝ ⎛ 5 8 − 17 16 ⎠ ⎞ , ⎝ ⎛ 2 11 − 21 17 ⎠ ⎞ ⎭ ⎬ ⎫ .
Question
b. find the coordinate vector [ x ] B [x]B [ x ] B
|-8|
|-58|
|151|
|-58|
Solution
b
x = P B [ x ] B → [ x ] B = P B − 1 x x = P _ {B} \left[ x \right] _ {B} \rightarrow \left[ x \right] _ {B} = P _ {B} ^ {- 1} x x = P B [ x ] B → [ x ] B = P B − 1 x x = ( − 8 − 58 151 − 58 ) , B = { ( 1 3 − 1 2 ) , ( 1 − 4 − 5 4 ) , ( 5 8 − 17 16 ) , ( 2 11 − 21 17 ) } → P B = ( 1 1 5 2 3 − 4 8 11 − 1 − 5 − 17 − 21 2 4 16 17 ) x = \left( \begin{array}{c} - 8 \\ - 5 8 \\ 1 5 1 \\ - 5 8 \end{array} \right), B = \left\{\left( \begin{array}{c} 1 \\ 3 \\ - 1 \\ 2 \end{array} \right), \left( \begin{array}{c} 1 \\ - 4 \\ - 5 \\ 4 \end{array} \right), \left( \begin{array}{c} 5 \\ 8 \\ - 1 7 \\ 1 6 \end{array} \right), \left( \begin{array}{c} 2 \\ 1 1 \\ - 2 1 \\ 1 7 \end{array} \right) \right\} \to P _ {B} = \left( \begin{array}{c c c c} 1 & 1 & 5 & 2 \\ 3 & - 4 & 8 & 1 1 \\ - 1 - 5 & - 1 7 & - 2 1 \\ 2 & 4 & 1 6 & 1 7 \end{array} \right) x = ⎝ ⎛ − 8 − 58 151 − 58 ⎠ ⎞ , B = ⎩ ⎨ ⎧ ⎝ ⎛ 1 3 − 1 2 ⎠ ⎞ , ⎝ ⎛ 1 − 4 − 5 4 ⎠ ⎞ , ⎝ ⎛ 5 8 − 17 16 ⎠ ⎞ , ⎝ ⎛ 2 11 − 21 17 ⎠ ⎞ ⎭ ⎬ ⎫ → P B = ⎝ ⎛ 1 3 − 1 − 5 2 1 − 4 − 17 4 5 8 − 21 16 2 11 17 ⎠ ⎞ det P B = ∣ 1 1 5 2 3 − 4 8 11 − 1 − 5 − 17 − 21 2 4 16 17 ∣ = { r 2 = r 2 + ( − 3 ) r 1 r 3 = r 3 + r 1 r 4 = r 4 + ( − 2 ) r 1 } = ∣ 1 1 5 2 0 − 7 − 7 5 0 − 4 − 12 − 19 0 2 6 13 ∣ = { r 3 = r 3 + ( − 4 7 ) r 2 r 4 = r 4 + ( 2 7 ) r 2 = ∣ 1 1 5 2 0 − 7 − 7 5 0 0 − 8 − 153 / 7 0 0 4 101 / 7 ∣ = { r 4 = r 4 + ( 1 2 ) r 3 } = ∣ 1 1 5 2 0 − 7 − 7 5 0 0 − 8 − 153 / 7 0 0 4 7 / 2 ∣ = 1 ⋅ ( − 7 ) ⋅ ( − 8 ) ⋅ 7 2 = 196 \begin{array}{l} \det P _ {B} = \left| \begin{array}{c c c c} 1 & 1 & 5 & 2 \\ 3 & - 4 & 8 & 1 1 \\ - 1 & - 5 & - 1 7 & - 2 1 \\ 2 & 4 & 1 6 & 1 7 \end{array} \right| = \left\{ \begin{array}{l} r 2 = r 2 + (- 3) r 1 \\ r 3 = r 3 + r 1 \\ r 4 = r 4 + (- 2) r 1 \end{array} \right\} = \left| \begin{array}{c c c c} 1 & 1 & 5 & 2 \\ 0 & - 7 & - 7 & 5 \\ 0 & - 4 & - 1 2 & - 1 9 \\ 0 & 2 & 6 & 1 3 \end{array} \right| = \left\{ \begin{array}{l} r 3 = r 3 + \left(\frac {- 4}{7}\right) r 2 \\ r 4 = r 4 + \left(\frac {2}{7}\right) r 2 \end{array} \right. \\ = \left| \begin{array}{c c c c} 1 & 1 & 5 & 2 \\ 0 & - 7 & - 7 & 5 \\ 0 & 0 & - 8 & - 1 5 3 / 7 \\ 0 & 0 & 4 & 1 0 1 / 7 \end{array} \right| = \left\{r 4 = r 4 + \left(\frac {1}{2}\right) r 3 \right\} = \left| \begin{array}{c c c c} 1 & 1 & 5 & 2 \\ 0 & - 7 & - 7 & 5 \\ 0 & 0 & - 8 & - 1 5 3 / 7 \\ 0 & 0 & 4 & 7 / 2 \end{array} \right| \\ = \frac {1 \cdot (- 7) \cdot (- 8) \cdot 7}{2} = 1 9 6 \\ \end{array} det P B = ∣ ∣ 1 3 − 1 2 1 − 4 − 5 4 5 8 − 17 16 2 11 − 21 17 ∣ ∣ = ⎩ ⎨ ⎧ r 2 = r 2 + ( − 3 ) r 1 r 3 = r 3 + r 1 r 4 = r 4 + ( − 2 ) r 1 ⎭ ⎬ ⎫ = ∣ ∣ 1 0 0 0 1 − 7 − 4 2 5 − 7 − 12 6 2 5 − 19 13 ∣ ∣ = { r 3 = r 3 + ( 7 − 4 ) r 2 r 4 = r 4 + ( 7 2 ) r 2 = ∣ ∣ 1 0 0 0 1 − 7 0 0 5 − 7 − 8 4 2 5 − 153/7 101/7 ∣ ∣ = { r 4 = r 4 + ( 2 1 ) r 3 } = ∣ ∣ 1 0 0 0 1 − 7 0 0 5 − 7 − 8 4 2 5 − 153/7 7/2 ∣ ∣ = 2 1 ⋅ ( − 7 ) ⋅ ( − 8 ) ⋅ 7 = 196 P 11 = ∣ − 4 8 11 − 5 − 17 − 21 4 16 17 ∣ = ( − 4 ) ⋅ ( − 17 ) ⋅ 17 + 8 ⋅ ( − 21 ) ⋅ 4 + 11 ⋅ ( − 5 ) ⋅ 16 − 11 ⋅ ( − 17 ) ⋅ 4 − ( − 4 ) ⋅ ( − 21 ) ⋅ 16 − 8 ⋅ ( − 5 ) ⋅ 17 = 1156 − 672 − 880 + 748 − 1344 + 680 = − 312 \begin{array}{l} P _ {1 1} = \left| \begin{array}{c c c} - 4 & 8 & 1 1 \\ - 5 & - 1 7 & - 2 1 \\ 4 & 1 6 & 1 7 \end{array} \right| \\ = (- 4) \cdot (- 1 7) \cdot 1 7 + 8 \cdot (- 2 1) \cdot 4 + 1 1 \cdot (- 5) \cdot 1 6 - 1 1 \cdot (- 1 7) \cdot 4 - (- 4) \\ \cdot (- 2 1) \cdot 1 6 - 8 \cdot (- 5) \cdot 1 7 = 1 1 5 6 - 6 7 2 - 8 8 0 + 7 4 8 - 1 3 4 4 + 6 8 0 = - 3 1 2 \\ \end{array} P 11 = ∣ ∣ − 4 − 5 4 8 − 17 16 11 − 21 17 ∣ ∣ = ( − 4 ) ⋅ ( − 17 ) ⋅ 17 + 8 ⋅ ( − 21 ) ⋅ 4 + 11 ⋅ ( − 5 ) ⋅ 16 − 11 ⋅ ( − 17 ) ⋅ 4 − ( − 4 ) ⋅ ( − 21 ) ⋅ 16 − 8 ⋅ ( − 5 ) ⋅ 17 = 1156 − 672 − 880 + 748 − 1344 + 680 = − 312 P 12 = ∣ 3 8 11 − 1 − 17 − 21 2 16 17 ∣ = 3 ⋅ ( − 17 ) ⋅ 17 + 8 ⋅ ( − 21 ) ⋅ 2 + 11 ⋅ ( − 1 ) ⋅ 16 − 11 ⋅ ( − 17 ) ⋅ 2 − 3 ⋅ ( − 21 ) ⋅ 16 − 8 ⋅ ( − 1 ) ⋅ 17 = − 867 − 336 − 176 + 374 + 1008 + 136 = 139 P_{12} = \left| \begin{array}{ccc} 3 & 8 & 11 \\ -1 & -17 & -21 \\ 2 & 16 & 17 \end{array} \right| = 3 \cdot (-17) \cdot 17 + 8 \cdot (-21) \cdot 2 + 11 \cdot (-1) \cdot 16 - 11 \cdot (-17) \cdot 2 - 3 \cdot (-21) \cdot 16 - 8 \cdot (-1) \cdot 17 = -867 - 336 - 176 + 374 + 1008 + 136 = 139 P 12 = ∣ ∣ 3 − 1 2 8 − 17 16 11 − 21 17 ∣ ∣ = 3 ⋅ ( − 17 ) ⋅ 17 + 8 ⋅ ( − 21 ) ⋅ 2 + 11 ⋅ ( − 1 ) ⋅ 16 − 11 ⋅ ( − 17 ) ⋅ 2 − 3 ⋅ ( − 21 ) ⋅ 16 − 8 ⋅ ( − 1 ) ⋅ 17 = − 867 − 336 − 176 + 374 + 1008 + 136 = 139 P 13 = ∣ 3 − 4 11 − 1 − 5 − 21 2 4 17 ∣ = 3 ⋅ ( − 5 ) ⋅ 17 + ( − 4 ) ⋅ ( − 21 ) ⋅ 2 + 11 ⋅ ( − 1 ) ⋅ 4 − 11 ⋅ ( − 5 ) ⋅ 2 − 3 ⋅ ( − 21 ) ⋅ 4 − ( − 4 ) ⋅ ( − 1 ) ⋅ 17 = − 255 + 168 − 44 + 110 + 252 − 68 = 163 P_{13} = \left| \begin{array}{ccc} 3 & -4 & 11 \\ -1 & -5 & -21 \\ 2 & 4 & 17 \end{array} \right| = 3 \cdot (-5) \cdot 17 + (-4) \cdot (-21) \cdot 2 + 11 \cdot (-1) \cdot 4 - 11 \cdot (-5) \cdot 2 - 3 \cdot (-21) \cdot 4 - (-4) \cdot (-1) \cdot 17 = -255 + 168 - 44 + 110 + 252 - 68 = 163 P 13 = ∣ ∣ 3 − 1 2 − 4 − 5 4 11 − 21 17 ∣ ∣ = 3 ⋅ ( − 5 ) ⋅ 17 + ( − 4 ) ⋅ ( − 21 ) ⋅ 2 + 11 ⋅ ( − 1 ) ⋅ 4 − 11 ⋅ ( − 5 ) ⋅ 2 − 3 ⋅ ( − 21 ) ⋅ 4 − ( − 4 ) ⋅ ( − 1 ) ⋅ 17 = − 255 + 168 − 44 + 110 + 252 − 68 = 163 P 14 = ∣ 3 − 4 8 − 1 − 5 − 17 2 4 16 ∣ = 3 ⋅ ( − 5 ) ⋅ 16 + ( − 4 ) ⋅ ( − 17 ) ⋅ 2 + 8 ⋅ ( − 1 ) ⋅ 4 − 8 ⋅ ( − 5 ) ⋅ 2 − 3 ⋅ ( − 17 ) ⋅ 4 − ( − 4 ) ⋅ ( − 1 ) ⋅ 16 = − 240 + 136 − 32 + 80 + 204 − 64 = 84 P_{14} = \left| \begin{array}{ccc} 3 & -4 & 8 \\ -1 & -5 & -17 \\ 2 & 4 & 16 \end{array} \right| = 3 \cdot (-5) \cdot 16 + (-4) \cdot (-17) \cdot 2 + 8 \cdot (-1) \cdot 4 - 8 \cdot (-5) \cdot 2 - 3 \cdot (-17) \cdot 4 - (-4) \cdot (-1) \cdot 16 = -240 + 136 - 32 + 80 + 204 - 64 = 84 P 14 = ∣ ∣ 3 − 1 2 − 4 − 5 4 8 − 17 16 ∣ ∣ = 3 ⋅ ( − 5 ) ⋅ 16 + ( − 4 ) ⋅ ( − 17 ) ⋅ 2 + 8 ⋅ ( − 1 ) ⋅ 4 − 8 ⋅ ( − 5 ) ⋅ 2 − 3 ⋅ ( − 17 ) ⋅ 4 − ( − 4 ) ⋅ ( − 1 ) ⋅ 16 = − 240 + 136 − 32 + 80 + 204 − 64 = 84 P 21 = ∣ 1 5 2 − 5 − 17 − 21 4 16 17 ∣ = 1 ⋅ ( − 17 ) ⋅ 17 + 5 ⋅ ( − 21 ) ⋅ 4 + 2 ⋅ ( − 5 ) ⋅ 16 − 2 ⋅ ( − 17 ) ⋅ 4 − 1 ⋅ ( − 21 ) ⋅ 16 − 5 ⋅ ( − 5 ) ⋅ 17 = − 289 − 420 − 160 + 136 + 336 + 425 = 28 P_{21} = \left| \begin{array}{ccc} 1 & 5 & 2 \\ -5 & -17 & -21 \\ 4 & 16 & 17 \end{array} \right| = 1 \cdot (-17) \cdot 17 + 5 \cdot (-21) \cdot 4 + 2 \cdot (-5) \cdot 16 - 2 \cdot (-17) \cdot 4 - 1 \cdot (-21) \cdot 16 - 5 \cdot (-5) \cdot 17 = -289 - 420 - 160 + 136 + 336 + 425 = 28 P 21 = ∣ ∣ 1 − 5 4 5 − 17 16 2 − 21 17 ∣ ∣ = 1 ⋅ ( − 17 ) ⋅ 17 + 5 ⋅ ( − 21 ) ⋅ 4 + 2 ⋅ ( − 5 ) ⋅ 16 − 2 ⋅ ( − 17 ) ⋅ 4 − 1 ⋅ ( − 21 ) ⋅ 16 − 5 ⋅ ( − 5 ) ⋅ 17 = − 289 − 420 − 160 + 136 + 336 + 425 = 28 P 22 = ∣ 1 5 2 − 1 − 17 − 21 2 16 17 ∣ = 1 ⋅ ( − 17 ) ⋅ 17 + 5 ⋅ ( − 21 ) ⋅ 2 + 2 ⋅ ( − 1 ) ⋅ 16 − 2 ⋅ ( − 17 ) ⋅ 2 − 1 ⋅ ( − 21 ) ⋅ 16 − 5 ⋅ ( − 1 ) ⋅ 17 = − 289 − 210 − 32 + 68 + 336 + 85 = − 42 P_{22} = \left| \begin{array}{ccc} 1 & 5 & 2 \\ -1 & -17 & -21 \\ 2 & 16 & 17 \end{array} \right| = 1 \cdot (-17) \cdot 17 + 5 \cdot (-21) \cdot 2 + 2 \cdot (-1) \cdot 16 - 2 \cdot (-17) \cdot 2 - 1 \cdot (-21) \cdot 16 - 5 \cdot (-1) \cdot 17 = -289 - 210 - 32 + 68 + 336 + 85 = -42 P 22 = ∣ ∣ 1 − 1 2 5 − 17 16 2 − 21 17 ∣ ∣ = 1 ⋅ ( − 17 ) ⋅ 17 + 5 ⋅ ( − 21 ) ⋅ 2 + 2 ⋅ ( − 1 ) ⋅ 16 − 2 ⋅ ( − 17 ) ⋅ 2 − 1 ⋅ ( − 21 ) ⋅ 16 − 5 ⋅ ( − 1 ) ⋅ 17 = − 289 − 210 − 32 + 68 + 336 + 85 = − 42 P 23 = ∣ 1 1 2 − 1 − 5 − 21 2 4 17 ∣ = 1 ⋅ ( − 5 ) ⋅ 17 + 1 ⋅ ( − 21 ) ⋅ 2 + 2 ⋅ ( − 1 ) ⋅ 4 − 2 ⋅ ( − 5 ) ⋅ 2 − 1 ⋅ ( − 21 ) ⋅ 4 − 1 ⋅ ( − 1 ) ⋅ 17 = − 85 − 42 − 8 + 20 + 84 + 17 = − 14 P_{23} = \left| \begin{array}{ccc} 1 & 1 & 2 \\ -1 & -5 & -21 \\ 2 & 4 & 17 \end{array} \right| = 1 \cdot (-5) \cdot 17 + 1 \cdot (-21) \cdot 2 + 2 \cdot (-1) \cdot 4 - 2 \cdot (-5) \cdot 2 - 1 \cdot (-21) \cdot 4 - 1 \cdot (-1) \cdot 17 = -85 - 42 - 8 + 20 + 84 + 17 = -14 P 23 = ∣ ∣ 1 − 1 2 1 − 5 4 2 − 21 17 ∣ ∣ = 1 ⋅ ( − 5 ) ⋅ 17 + 1 ⋅ ( − 21 ) ⋅ 2 + 2 ⋅ ( − 1 ) ⋅ 4 − 2 ⋅ ( − 5 ) ⋅ 2 − 1 ⋅ ( − 21 ) ⋅ 4 − 1 ⋅ ( − 1 ) ⋅ 17 = − 85 − 42 − 8 + 20 + 84 + 17 = − 14 P 24 = ∣ 1 1 5 − 1 − 5 − 17 2 4 16 ∣ = 1 ⋅ ( − 5 ) ⋅ 16 + 1 ⋅ ( − 17 ) ⋅ 2 + 5 ⋅ ( − 1 ) ⋅ 4 − 5 ⋅ ( − 5 ) ⋅ 2 − 1 ⋅ ( − 17 ) ⋅ 4 − 1 ⋅ ( − 1 ) ⋅ 16 = − 80 − 34 − 20 + 50 + 68 + 16 = 0 P_{24} = \left| \begin{array}{ccc} 1 & 1 & 5 \\ -1 & -5 & -17 \\ 2 & 4 & 16 \end{array} \right| = 1 \cdot (-5) \cdot 16 + 1 \cdot (-17) \cdot 2 + 5 \cdot (-1) \cdot 4 - 5 \cdot (-5) \cdot 2 - 1 \cdot (-17) \cdot 4 - 1 \cdot (-1) \cdot 16 = -80 - 34 - 20 + 50 + 68 + 16 = 0 P 24 = ∣ ∣ 1 − 1 2 1 − 5 4 5 − 17 16 ∣ ∣ = 1 ⋅ ( − 5 ) ⋅ 16 + 1 ⋅ ( − 17 ) ⋅ 2 + 5 ⋅ ( − 1 ) ⋅ 4 − 5 ⋅ ( − 5 ) ⋅ 2 − 1 ⋅ ( − 17 ) ⋅ 4 − 1 ⋅ ( − 1 ) ⋅ 16 = − 80 − 34 − 20 + 50 + 68 + 16 = 0 P 31 = ∣ 1 5 2 − 4 8 11 4 16 17 ∣ = 1 ⋅ 8 ⋅ 17 + 5 ⋅ 11 ⋅ 4 + 2 ⋅ ( − 4 ) ⋅ 16 − 2 ⋅ 8 ⋅ 4 − 1 ⋅ 11 ⋅ 16 − 5 ⋅ ( − 4 ) ⋅ 17 = 136 + 220 − 128 − 64 − 176 + 340 = 328 P_{31} = \left| \begin{array}{ccc} 1 & 5 & 2 \\ -4 & 8 & 11 \\ 4 & 16 & 17 \end{array} \right| = 1 \cdot 8 \cdot 17 + 5 \cdot 11 \cdot 4 + 2 \cdot (-4) \cdot 16 - 2 \cdot 8 \cdot 4 - 1 \cdot 11 \cdot 16 - 5 \cdot (-4) \cdot 17 = 136 + 220 - 128 - 64 - 176 + 340 = 328 P 31 = ∣ ∣ 1 − 4 4 5 8 16 2 11 17 ∣ ∣ = 1 ⋅ 8 ⋅ 17 + 5 ⋅ 11 ⋅ 4 + 2 ⋅ ( − 4 ) ⋅ 16 − 2 ⋅ 8 ⋅ 4 − 1 ⋅ 11 ⋅ 16 − 5 ⋅ ( − 4 ) ⋅ 17 = 136 + 220 − 128 − 64 − 176 + 340 = 328 P 32 = ∣ 1 5 2 3 8 11 2 16 17 ∣ = 1 ⋅ 8 ⋅ 17 + 5 ⋅ 11 ⋅ 2 + 2 ⋅ 3 ⋅ 16 − 2 ⋅ 8 ⋅ 2 − 1 ⋅ 11 ⋅ 16 − 5 ⋅ 3 ⋅ 17 P_{32} = \left| \begin{array}{ccc} 1 & 5 & 2 \\ 3 & 8 & 11 \\ 2 & 16 & 17 \end{array} \right| = 1 \cdot 8 \cdot 17 + 5 \cdot 11 \cdot 2 + 2 \cdot 3 \cdot 16 - 2 \cdot 8 \cdot 2 - 1 \cdot 11 \cdot 16 - 5 \cdot 3 \cdot 17 P 32 = ∣ ∣ 1 3 2 5 8 16 2 11 17 ∣ ∣ = 1 ⋅ 8 ⋅ 17 + 5 ⋅ 11 ⋅ 2 + 2 ⋅ 3 ⋅ 16 − 2 ⋅ 8 ⋅ 2 − 1 ⋅ 11 ⋅ 16 − 5 ⋅ 3 ⋅ 17 = 136 + 110 + 96 − 32 − 176 − 255 = − 121 = 136 + 110 + 96 - 32 - 176 - 255 = -121 = 136 + 110 + 96 − 32 − 176 − 255 = − 121 P 33 = ∣ 1 1 2 3 − 4 11 2 4 17 ∣ = 1 ⋅ ( − 4 ) ⋅ 17 + 1 ⋅ 11 ⋅ 2 + 2 ⋅ 3 ⋅ 4 − 2 ⋅ ( − 4 ) ⋅ 2 − 1 ⋅ 11 ⋅ 4 − 1 ⋅ 3 ⋅ 17 P_{33} = \left| \begin{array}{ccc} 1 & 1 & 2 \\ 3 & -4 & 11 \\ 2 & 4 & 17 \end{array} \right| = 1 \cdot (-4) \cdot 17 + 1 \cdot 11 \cdot 2 + 2 \cdot 3 \cdot 4 - 2 \cdot (-4) \cdot 2 - 1 \cdot 11 \cdot 4 - 1 \cdot 3 \cdot 17 P 33 = ∣ ∣ 1 3 2 1 − 4 4 2 11 17 ∣ ∣ = 1 ⋅ ( − 4 ) ⋅ 17 + 1 ⋅ 11 ⋅ 2 + 2 ⋅ 3 ⋅ 4 − 2 ⋅ ( − 4 ) ⋅ 2 − 1 ⋅ 11 ⋅ 4 − 1 ⋅ 3 ⋅ 17 = − 68 + 22 + 24 + 16 − 44 − 51 = − 101 = -68 + 22 + 24 + 16 - 44 - 51 = -101 = − 68 + 22 + 24 + 16 − 44 − 51 = − 101 P 34 = ∣ 1 1 5 3 − 4 8 2 4 16 ∣ = 1 ⋅ ( − 4 ) ⋅ 16 + 1 ⋅ 8 ⋅ 2 + 5 ⋅ 3 ⋅ 4 − 5 ⋅ ( − 4 ) ⋅ 2 − 1 ⋅ 8 ⋅ 4 − 1 ⋅ 3 ⋅ 16 P_{34} = \left| \begin{array}{ccc} 1 & 1 & 5 \\ 3 & -4 & 8 \\ 2 & 4 & 16 \end{array} \right| = 1 \cdot (-4) \cdot 16 + 1 \cdot 8 \cdot 2 + 5 \cdot 3 \cdot 4 - 5 \cdot (-4) \cdot 2 - 1 \cdot 8 \cdot 4 - 1 \cdot 3 \cdot 16 P 34 = ∣ ∣ 1 3 2 1 − 4 4 5 8 16 ∣ ∣ = 1 ⋅ ( − 4 ) ⋅ 16 + 1 ⋅ 8 ⋅ 2 + 5 ⋅ 3 ⋅ 4 − 5 ⋅ ( − 4 ) ⋅ 2 − 1 ⋅ 8 ⋅ 4 − 1 ⋅ 3 ⋅ 16 = − 64 + 16 + 60 + 40 − 32 − 48 = − 28 = -64 + 16 + 60 + 40 - 32 - 48 = -28 = − 64 + 16 + 60 + 40 − 32 − 48 = − 28 P 41 = ∣ 1 5 2 − 4 8 11 − 5 − 17 − 21 ∣ P_{41} = \left| \begin{array}{ccc} 1 & 5 & 2 \\ -4 & 8 & 11 \\ -5 & -17 & -21 \end{array} \right| P 41 = ∣ ∣ 1 − 4 − 5 5 8 − 17 2 11 − 21 ∣ ∣ = 1 ⋅ 8 ⋅ ( − 21 ) + 5 ⋅ 11 ⋅ ( − 5 ) + 2 ⋅ ( − 4 ) ⋅ ( − 17 ) − 2 ⋅ 8 ⋅ ( − 5 ) − 1 ⋅ 11 ⋅ ( − 17 ) = 1 \cdot 8 \cdot (-21) + 5 \cdot 11 \cdot (-5) + 2 \cdot (-4) \cdot (-17) - 2 \cdot 8 \cdot (-5) - 1 \cdot 11 \cdot (-17) = 1 ⋅ 8 ⋅ ( − 21 ) + 5 ⋅ 11 ⋅ ( − 5 ) + 2 ⋅ ( − 4 ) ⋅ ( − 17 ) − 2 ⋅ 8 ⋅ ( − 5 ) − 1 ⋅ 11 ⋅ ( − 17 ) − 5 ⋅ ( − 4 ) ⋅ ( − 21 ) = − 168 − 275 + 136 + 80 + 187 − 420 = − 460 -5 \cdot (-4) \cdot (-21) = -168 - 275 + 136 + 80 + 187 - 420 = -460 − 5 ⋅ ( − 4 ) ⋅ ( − 21 ) = − 168 − 275 + 136 + 80 + 187 − 420 = − 460 P 42 = ∣ 1 5 2 3 8 11 − 1 − 17 − 21 ∣ P_{42} = \left| \begin{array}{ccc} 1 & 5 & 2 \\ 3 & 8 & 11 \\ -1 & -17 & -21 \end{array} \right| P 42 = ∣ ∣ 1 3 − 1 5 8 − 17 2 11 − 21 ∣ ∣ = 1 ⋅ 8 ⋅ ( − 21 ) + 5 ⋅ 11 ⋅ ( − 1 ) + 2 ⋅ 3 ⋅ ( − 17 ) − 2 ⋅ 8 ⋅ ( − 1 ) − 1 ⋅ 11 ⋅ ( − 17 ) − 5 = 1 \cdot 8 \cdot (-21) + 5 \cdot 11 \cdot (-1) + 2 \cdot 3 \cdot (-17) - 2 \cdot 8 \cdot (-1) - 1 \cdot 11 \cdot (-17) - 5 = 1 ⋅ 8 ⋅ ( − 21 ) + 5 ⋅ 11 ⋅ ( − 1 ) + 2 ⋅ 3 ⋅ ( − 17 ) − 2 ⋅ 8 ⋅ ( − 1 ) − 1 ⋅ 11 ⋅ ( − 17 ) − 5 ⋅ 3 ⋅ ( − 21 ) = − 168 − 55 − 102 + 16 + 187 + 315 = 193 \cdot 3 \cdot (-21) = -168 - 55 - 102 + 16 + 187 + 315 = 193 ⋅ 3 ⋅ ( − 21 ) = − 168 − 55 − 102 + 16 + 187 + 315 = 193 P 43 = ∣ 1 1 2 3 − 4 11 − 1 − 5 − 21 ∣ P_{43} = \left| \begin{array}{ccc} 1 & 1 & 2 \\ 3 & -4 & 11 \\ -1 & -5 & -21 \end{array} \right| P 43 = ∣ ∣ 1 3 − 1 1 − 4 − 5 2 11 − 21 ∣ ∣ = 1 ⋅ ( − 4 ) ⋅ ( − 21 ) + 1 ⋅ 11 ⋅ ( − 1 ) + 2 ⋅ 3 ⋅ ( − 5 ) − 2 ⋅ ( − 4 ) ⋅ ( − 1 ) − 1 ⋅ 11 ⋅ ( − 5 ) = 1 \cdot (-4) \cdot (-21) + 1 \cdot 11 \cdot (-1) + 2 \cdot 3 \cdot (-5) - 2 \cdot (-4) \cdot (-1) - 1 \cdot 11 \cdot (-5) = 1 ⋅ ( − 4 ) ⋅ ( − 21 ) + 1 ⋅ 11 ⋅ ( − 1 ) + 2 ⋅ 3 ⋅ ( − 5 ) − 2 ⋅ ( − 4 ) ⋅ ( − 1 ) − 1 ⋅ 11 ⋅ ( − 5 ) − 1 ⋅ 3 ⋅ ( − 21 ) = 84 − 11 − 30 − 8 + 55 + 63 = 153 -1 \cdot 3 \cdot (-21) = 84 - 11 - 30 - 8 + 55 + 63 = 153 − 1 ⋅ 3 ⋅ ( − 21 ) = 84 − 11 − 30 − 8 + 55 + 63 = 153 P 44 = ∣ 1 1 5 3 − 4 8 − 1 − 5 − 17 ∣ P_{44} = \left| \begin{array}{ccc} 1 & 1 & 5 \\ 3 & -4 & 8 \\ -1 & -5 & -17 \end{array} \right| P 44 = ∣ ∣ 1 3 − 1 1 − 4 − 5 5 8 − 17 ∣ ∣ = 1 ⋅ ( − 4 ) ⋅ ( − 17 ) + 1 ⋅ 8 ⋅ ( − 1 ) + 5 ⋅ 3 ⋅ ( − 5 ) − 5 ⋅ ( − 4 ) ⋅ ( − 1 ) − 1 ⋅ 8 ⋅ ( − 5 ) = 1 \cdot (-4) \cdot (-17) + 1 \cdot 8 \cdot (-1) + 5 \cdot 3 \cdot (-5) - 5 \cdot (-4) \cdot (-1) - 1 \cdot 8 \cdot (-5) = 1 ⋅ ( − 4 ) ⋅ ( − 17 ) + 1 ⋅ 8 ⋅ ( − 1 ) + 5 ⋅ 3 ⋅ ( − 5 ) − 5 ⋅ ( − 4 ) ⋅ ( − 1 ) − 1 ⋅ 8 ⋅ ( − 5 ) − 1 ⋅ 3 ⋅ ( − 17 ) = 68 − 8 − 75 − 20 + 40 + 51 = 56 -1 \cdot 3 \cdot (-17) = 68 - 8 - 75 - 20 + 40 + 51 = 56 − 1 ⋅ 3 ⋅ ( − 17 ) = 68 − 8 − 75 − 20 + 40 + 51 = 56 P ∗ = ( P 11 − P 12 P 13 − P 14 − P 21 P 22 − P 23 P 24 P 31 − P 32 P 33 − P 34 − P 41 P 42 − P 43 P 44 ) = ( − 312 − 139 163 − 84 − 28 − 42 14 0 328 121 − 101 28 460 193 − 153 56 ) P^{*} = \left( \begin{array}{rrr} P_{11} & -P_{12} & P_{13} & -P_{14} \\ -P_{21} & P_{22} & -P_{23} & P_{24} \\ P_{31} & -P_{32} & P_{33} & -P_{34} \\ -P_{41} & P_{42} & -P_{43} & P_{44} \end{array} \right) = \left( \begin{array}{rrr} -312 - 139 & 163 & -84 \\ -28 & -42 & 14 & 0 \\ 328 & 121 & -101 & 28 \\ 460 & 193 & -153 & 56 \end{array} \right) P ∗ = ⎝ ⎛ P 11 − P 21 P 31 − P 41 − P 12 P 22 − P 32 P 42 P 13 − P 23 P 33 − P 43 − P 14 P 24 − P 34 P 44 ⎠ ⎞ = ⎝ ⎛ − 312 − 139 − 28 328 460 163 − 42 121 193 − 84 14 − 101 − 153 0 28 56 ⎠ ⎞ P ∗ T = ( − 312 − 28 328 460 − 139 − 42 121 193 163 14 − 101 − 153 − 84 0 28 56 ) P^{*T} = \left( \begin{array}{rrr} -312 - 28 & 328 & 460 \\ -139 - 42 & 121 & 193 \\ 163 & 14 & -101 - 153 \\ -84 & 0 & 28 & 56 \end{array} \right) P ∗ T = ⎝ ⎛ − 312 − 28 − 139 − 42 163 − 84 328 121 14 0 460 193 − 101 − 153 28 56 ⎠ ⎞ P B − 1 = P ∗ T d e t P B = ( − 78 / 49 − 1 / 7 82 / 49 115 / 49 − 139 / 196 − 3 / 14 121 / 196 193 / 196 163 / 196 1 / 14 − 101 / 196 − 153 / 196 − 3 / 7 0 1 / 7 2 / 7 ) P _ {B} ^ {- 1} = \frac {P ^ {* T}}{d e t P _ {B}} = \left( \begin{array}{c c c c} - 7 8 / 4 9 & - 1 / 7 & 8 2 / 4 9 & 1 1 5 / 4 9 \\ - 1 3 9 / 1 9 6 & - 3 / 1 4 & 1 2 1 / 1 9 6 & 1 9 3 / 1 9 6 \\ 1 6 3 / 1 9 6 & 1 / 1 4 & - 1 0 1 / 1 9 6 & - 1 5 3 / 1 9 6 \\ - 3 / 7 & 0 & 1 / 7 & 2 / 7 \end{array} \right) P B − 1 = d e t P B P ∗ T = ⎝ ⎛ − 78/49 − 139/196 163/196 − 3/7 − 1/7 − 3/14 1/14 0 82/49 121/196 − 101/196 1/7 115/49 193/196 − 153/196 2/7 ⎠ ⎞ [ x ] B = P B − 1 x = − 8 ( − 78 49 − 139 196 163 196 − 3 7 ) − 58 ( − 1 7 − 3 14 1 14 0 ) + 151 ( 82 49 121 196 − 101 196 1 7 ) − 58 ( 115 49 193 196 − 153 196 2 7 ) = ( 6742 49 10625 196 − 8493 196 59 7 ) . [ x ] _ {B} = P _ {B} ^ {- 1} x = - 8 \left( \begin{array}{c} - \frac {7 8}{4 9} \\ - \frac {1 3 9}{1 9 6} \\ \frac {1 6 3}{1 9 6} \\ - \frac {3}{7} \end{array} \right) - 5 8 \left( \begin{array}{c} - \frac {1}{7} \\ - \frac {3}{1 4} \\ \frac {1}{1 4} \\ 0 \end{array} \right) + 1 5 1 \left( \begin{array}{c} \frac {8 2}{4 9} \\ \frac {1 2 1}{1 9 6} \\ - \frac {1 0 1}{1 9 6} \\ \frac {1}{7} \end{array} \right) - 5 8 \left( \begin{array}{c} \frac {1 1 5}{4 9} \\ \frac {1 9 3}{1 9 6} \\ - \frac {1 5 3}{1 9 6} \\ \frac {2}{7} \end{array} \right) = \left( \begin{array}{c} \frac {6 7 4 2}{4 9} \\ \frac {1 0 6 2 5}{1 9 6} \\ - \frac {8 4 9 3}{1 9 6} \\ \frac {5 9}{7} \end{array} \right). [ x ] B = P B − 1 x = − 8 ⎝ ⎛ − 49 78 − 196 139 196 163 − 7 3 ⎠ ⎞ − 58 ⎝ ⎛ − 7 1 − 14 3 14 1 0 ⎠ ⎞ + 151 ⎝ ⎛ 49 82 196 121 − 196 101 7 1 ⎠ ⎞ − 58 ⎝ ⎛ 49 115 196 193 − 196 153 7 2 ⎠ ⎞ = ⎝ ⎛ 49 6742 196 10625 − 196 8493 7 59 ⎠ ⎞ .
Answer: [ x ] B = ( 6742 49 10625 196 − 8493 196 59 7 ) . [x]_B = \left( \begin{array}{c} \frac{6742}{49} \\ \frac{10625}{196} \\ -\frac{8493}{196} \\ \frac{59}{7} \end{array} \right). [ x ] B = ⎝ ⎛ 49 6742 196 10625 − 196 8493 7 59 ⎠ ⎞ .
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#340153 on Dec 2023