單元13 基礎題類題

  1. 已知方程組\(L:\left\{ \begin{array}{l}3x - y - 2z = - 5\\x - y + 2z = 13\\2x + y + z = 5\end{array} \right.\),則:
    (1) \(L\)之係數矩陣為__________。
    (2) \(L\)之增廣矩陣為__________。
    (3) 利用\(L\)之增廣矩陣進行列運算可得\(L\)之解為__________。
  1. __________設三元一次聯立方程式\(L\)之增廣矩陣為\(M\),請選出正確的選項。(多選)
    (1) 若\(M\)經矩陣列運算可化為\(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}2&1&3&5\\0&1&7&7\\0&0&1&1\end{array}{\kern 1pt} } \right]\),則\(L\)有無限多組解
    (2) 若\(M\)經矩陣列運算可化為\(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}2&1&3&5\\0&1&7&7\\0&0&0&1\end{array}{\kern 1pt} } \right]\),則\(L\)有無限多組解
    (3) 若\(M\)經矩陣列運算可化為\(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}2&1&3&5\\0&1&7&7\\0&0&0&0\end{array}{\kern 1pt} } \right]\),則\(L\)有無限多組解
    (4) 若\(M\)經矩陣列運算可化為\(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}2&1&3&5\\4&2&6&{10}\\0&0&1&1\end{array}{\kern 1pt} } \right]\),則\(L\)有無限多組解
    (5) 若\(M\)經矩陣列運算可化為\(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}2&1&3&5\\4&2&6&7\\0&0&1&1\end{array}{\kern 1pt} } \right]\),則\(L\)有無限多組解
  1. 下列算式為一個關於\(x\),\(y\),\(z\)方程組的矩陣列運算:
    \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}5&2&3&{ - 1}\\1&1&1&0\\2&{ - 1}&2&3\end{array}{\kern 1pt} } \right]\)\( \to \)\(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}1&a&1&0\\5&2&3&{ - 1}\\2&{ - 1}&2&3\end{array}{\kern 1pt} } \right]\)\( \to \)\(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}1&a&1&0\\0&{ - 3}&b&{ - 1}\\0&{ - 3}&0&3\end{array}{\kern 1pt} } \right]\)\( \to \)\(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}1&a&1&0\\0&{ - 3}&b&{ - 1}\\0&0&2&c\end{array}{\kern 1pt} } \right]\)\( \to \)\( \cdots \)
    若繼續進行列運算,即可求得\(x\),\(y\),\(z\)之解,則式中序組\((\,a\;,\;b\;,\;c\,) = \)__________。
  1. 設\(A = \left[ {{\kern 1pt} \begin{array}{*{20}{c}}2&1&3\\3&1&2\end{array}{\kern 1pt} } \right]\),\(B = \left[ {{\kern 1pt} \begin{array}{*{20}{c}}{ - 2}&{ - 5}&3\\1&{ - 3}&2\end{array}{\kern 1pt} } \right]\),則\(2A + B = \)__________。
  1. 設\(A = \left[ {{\kern 1pt} \begin{array}{*{20}{c}}1&2\\3&6\end{array}{\kern 1pt} } \right]\),\(B = \left[ {{\kern 1pt} \begin{array}{*{20}{c}}{ - 2}&{ - 4}\\1&2\end{array}{\kern 1pt} } \right]\),\(C = \left[ {{\kern 1pt} \begin{array}{*{20}{c}}0&6\\0&{ - 3}\end{array}{\kern 1pt} } \right]\),則:
    (1) \(AB = \)__________,\(BA = \)__________。
    (2) \(AC = \)__________。
  2. 已知\(A\)為二階方陣,且滿足\(A\left[ {{\kern 1pt} \begin{array}{*{20}{c}}2\\3\end{array}{\kern 1pt} } \right] = \left[ {{\kern 1pt} \begin{array}{*{20}{c}}1\\7\end{array}{\kern 1pt} } \right]\),\(A\left[ {{\kern 1pt} \begin{array}{*{20}{c}}3\\5\end{array}{\kern 1pt} } \right] = \left[ {{\kern 1pt} \begin{array}{*{20}{c}}4\\9\end{array}{\kern 1pt} } \right]\),則
    \(A\left[ {{\kern 1pt} \begin{array}{*{20}{c}}2&3\\3&5\end{array}{\kern 1pt} } \right] = \)__________。
  1. 設矩陣\(A = \left[ {{\kern 1pt} \begin{array}{*{20}{c}}2&3\\3&5\end{array}{\kern 1pt} } \right]\),請回答下列問題:
    (1) 矩陣\(A\)之乘法反方陣\({A^{ - 1}} = \)__________。
    (2) 若矩陣\(B\)之乘法反方陣\({B^{ - 1}} = \left[ {{\kern 1pt} \begin{array}{*{20}{c}}2&3\\3&5\end{array}{\kern 1pt} } \right]\),則\(B = \)__________。
    (3) 若矩陣\(C\)滿足\(C\left[ {{\kern 1pt} \begin{array}{*{20}{c}}2&3\\3&5\end{array}{\kern 1pt} } \right] = \left[ {{\kern 1pt} \begin{array}{*{20}{c}}1&4\\7&9\end{array}{\kern 1pt} } \right]\),則\(C = \)__________。
    (4) 若矩陣\(X\)滿足\(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}2&3\\3&5\end{array}{\kern 1pt} } \right]X = \left[ {{\kern 1pt} \begin{array}{*{20}{c}}1\\2\end{array}{\kern 1pt} } \right]\),則\(X = \)__________。
    (5) 二元一次聯立方程式\(\left\{ \begin{array}{l}2x + 3y = 1\\3x + 5y = 2\end{array} \right.\)的解為\(x = \)_____,\(y = \)_____。
  1. 下列哪些為轉移矩陣?(多選)
    (1) \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}{0.1}&{0.8}\\{0.5}&{0.3}\end{array}{\kern 1pt} } \right]\) (2) \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}{0.5}&{0.5}\\{0.5}&{0.5}\end{array}{\kern 1pt} } \right]\)  (3) \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}{0.2}&{0.7}\\{0.8}&{0.3}\end{array}{\kern 1pt} } \right]\)  (4) \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}{0.2}&{0.1}\\{1.3}&{0.5}\end{array}{\kern 1pt} } \right]\)
    (5) \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}{0.3}&{ - 0.8}\\{0.7}&{1.8}\end{array}{\kern 1pt} } \right]\)
  1. 某縣縣政府每週五對全縣居民發放甲、乙兩種彩券,每位居民均可憑身分證免費選擇領取甲券一張或乙券一張。根據長期統計,上週選擇甲券的民眾會有85%在本週維持選擇甲券、15%改選乙券;而選擇乙券的民眾會有35%在本週改選甲券、65%維持乙券。已知一開始時,領取甲券及乙券的民眾比例各為50%,而所謂穩定狀態,係指領取甲券及乙券的民眾比例在每週均保持不變。
    (1) 1年後,選擇甲券的民眾比例為__________。
    (2) 2年後,選擇甲券的民眾比例為__________。
    (3) 當領取甲券及乙券的民眾比例形成穩定狀態,此時領取甲券的民眾比
    例為__________。

Ans:

  1. (1) \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}3&{ - 1}&{ - 2}\\1&{ - 1}&2\\2&1&1\end{array}{\kern 1pt} } \right]\) (2) \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}3&{ - 1}&{ - 2}&{ - 5}\\1&{ - 1}&2&{13}\\2&1&1&5\end{array}{\kern 1pt} } \right]\) (3) \((\,x\;,\;y\;,\;z\,) = (\,1\;,\; - 2\;,\;5\,)\)
  2. (3)(4)
  3. \((\,1\;,\; - 2\;,\;4\,)\)
  4. \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}2&{ - 3}&9\\7&{ - 1}&6\end{array}{\kern 1pt} } \right]\)
  5. (1) \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}0&0\\0&0\end{array}{\kern 1pt} } \right]\),\(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}{ - 14}&{ - 28}\\7&{14}\end{array}{\kern 1pt} } \right]\)(2) \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}0&0\\0&0\end{array}{\kern 1pt} } \right]\)
  6. \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}1&4\\7&9\end{array}{\kern 1pt} } \right]\)
  7. (1) \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}5&{ - 3}\\{ - 3}&2\end{array}{\kern 1pt} } \right]\) (2) \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}5&{ - 3}\\{ - 3}&2\end{array}{\kern 1pt} } \right]\) (3) \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}{ - 7}&5\\8&{ - 3}\end{array}{\kern 1pt} } \right]\)(4) \(\left[ {{\kern 1pt} \begin{array}{*{20}{c}}{ - 1}\\1\end{array}{\kern 1pt} } \right]\) (5) \( - 1\),\(1\)
  8. (2)(3)
  9. (1) 0.6 (2) 0.65 (3) 0.7

 

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