《高等代數(shù)》題庫及答案

一、單項選擇題（每題2分，共10題）1.在實數(shù)域上，下列哪個矩陣是可逆的？A.\(\begin{pmatrix}1&2\\2&4\end{pmatrix}\)B.\(\begin{pmatrix}1&2\\3&4\end{pmatrix}\)C.\(\begin{pmatrix}0&1\\1&0\end{pmatrix}\)D.\(\begin{pmatrix}2&3\\4&6\end{pmatrix}\)答案：B2.多項式\(f(x)=x^3-3x+2\)在實數(shù)域上的根的個數(shù)是：A.0B.1C.2D.3答案：D3.下列哪個向量是線性無關(guān)的？A.\(\begin{pmatrix}1\\2\\3\end{pmatrix}\),\(\begin{pmatrix}2\\4\\6\end{pmatrix}\),\(\begin{pmatrix}3\\6\\9\end{pmatrix}\)B.\(\begin{pmatrix}1\\0\\0\end{pmatrix}\),\(\begin{pmatrix}0\\1\\0\end{pmatrix}\),\(\begin{pmatrix}0\\0\\1\end{pmatrix}\)C.\(\begin{pmatrix}1\\1\\1\end{pmatrix}\),\(\begin{pmatrix}1\\2\\3\end{pmatrix}\),\(\begin{pmatrix}2\\3\\4\end{pmatrix}\)D.\(\begin{pmatrix}1\\2\\3\end{pmatrix}\),\(\begin{pmatrix}4\\5\\6\end{pmatrix}\),\(\begin{pmatrix}7\\8\\9\end{pmatrix}\)答案：B4.下列哪個矩陣是正定矩陣？A.\(\begin{pmatrix}1&2\\2&1\end{pmatrix}\)B.\(\begin{pmatrix}1&-2\\-2&1\end{pmatrix}\)C.\(\begin{pmatrix}2&3\\3&2\end{pmatrix}\)D.\(\begin{pmatrix}-1&2\\2&-1\end{pmatrix}\)答案：C5.下列哪個向量是線性組合？A.\(\begin{pmatrix}1\\2\end{pmatrix}\),\(\begin{pmatrix}3\\4\end{pmatrix}\),\(\begin{pmatrix}5\\6\end{pmatrix}\)B.\(\begin{pmatrix}1\\0\end{pmatrix}\),\(\begin{pmatrix}0\\1\end{pmatrix}\),\(\begin{pmatrix}1\\1\end{pmatrix}\)C.\(\begin{pmatrix}2\\3\end{pmatrix}\),\(\begin{pmatrix}4\\5\end{pmatrix}\),\(\begin{pmatrix}6\\7\end{pmatrix}\)D.\(\begin{pmatrix}1\\2\end{pmatrix}\),\(\begin{pmatrix}3\\4\end{pmatrix}\),\(\begin{pmatrix}5\\6\end{pmatrix}\)答案：B6.下列哪個矩陣是上三角矩陣？A.\(\begin{pmatrix}1&2\\3&4\end{pmatrix}\)B.\(\begin{pmatrix}1&0&0\\0&2&0\\0&0&3\end{pmatrix}\)C.\(\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}\)D.\(\begin{pmatrix}1&2\\0&3\end{pmatrix}\)答案：B7.下列哪個矩陣是下三角矩陣？A.\(\begin{pmatrix}1&2\\3&4\end{pmatrix}\)B.\(\begin{pmatrix}1&0&0\\0&2&0\\0&0&3\end{pmatrix}\)C.\(\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}\)D.\(\begin{pmatrix}1&0\\2&3\end{pmatrix}\)答案：D8.下列哪個矩陣是單位矩陣？A.\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\)B.\(\begin{pmatrix}2&0\\0&2\end{pmatrix}\)C.\(\begin{pmatrix}1&1\\1&1\end{pmatrix}\)D.\(\begin{pmatrix}0&1\\1&0\end{pmatrix}\)答案：A9.下列哪個矩陣是零矩陣？A.\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\)B.\(\begin{pmatrix}0&0\\0&0\end{pmatrix}\)C.\(\begin{pmatrix}2&3\\4&5\end{pmatrix}\)D.\(\begin{pmatrix}1&2\\3&4\end{pmatrix}\)答案：B10.下列哪個矩陣是正交矩陣？A.\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\)B.\(\begin{pmatrix}0&1\\1&0\end{pmatrix}\)C.\(\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\end{pmatrix}\)D.\(\begin{pmatrix}1&1\\1&-1\end{pmatrix}\)答案：C二、多項選擇題（每題2分，共10題）1.下列哪些矩陣是可逆的？A.\(\begin{pmatrix}1&2\\2&4\end{pmatrix}\)B.\(\begin{pmatrix}1&2\\3&4\end{pmatrix}\)C.\(\begin{pmatrix}0&1\\1&0\end{pmatrix}\)D.\(\begin{pmatrix}2&3\\4&6\end{pmatrix}\)答案：B,C2.下列哪些向量是線性無關(guān)的？A.\(\begin{pmatrix}1\\2\\3\end{pmatrix}\),\(\begin{pmatrix}2\\4\\6\end{pmatrix}\),\(\begin{pmatrix}3\\6\\9\end{pmatrix}\)B.\(\begin{pmatrix}1\\0\\0\end{pmatrix}\),\(\begin{pmatrix}0\\1\\0\end{pmatrix}\),\(\begin{pmatrix}0\\0\\1\end{pmatrix}\)C.\(\begin{pmatrix}1\\1\\1\end{pmatrix}\),\(\begin{pmatrix}1\\2\\3\end{pmatrix}\),\(\begin{pmatrix}2\\3\\4\end{pmatrix}\)D.\(\begin{pmatrix}1\\2\\3\end{pmatrix}\),\(\begin{pmatrix}4\\5\\6\end{pmatrix}\),\(\begin{pmatrix}7\\8\\9\end{pmatrix}\)答案：B,C3.下列哪些矩陣是正定矩陣？A.\(\begin{pmatrix}1&2\\2&1\end{pmatrix}\)B.\(\begin{pmatrix}1&-2\\-2&1\end{pmatrix}\)C.\(\begin{pmatrix}2&3\\3&2\end{pmatrix}\)D.\(\begin{pmatrix}-1&2\\2&-1\end{pmatrix}\)答案：C4.下列哪些向量是線性組合？A.\(\begin{pmatrix}1\\2\end{pmatrix}\),\(\begin{pmatrix}3\\4\end{pmatrix}\),\(\begin{pmatrix}5\\6\end{pmatrix}\)B.\(\begin{pmatrix}1\\0\end{pmatrix}\),\(\begin{pmatrix}0\\1\end{pmatrix}\),\(\begin{pmatrix}1\\1\end{pmatrix}\)C.\(\begin{pmatrix}2\\3\end{pmatrix}\),\(\begin{pmatrix}4\\5\end{pmatrix}\),\(\begin{pmatrix}6\\7\end{pmatrix}\)D.\(\begin{pmatrix}1\\2\end{pmatrix}\),\(\begin{pmatrix}3\\4\end{pmatrix}\),\(\begin{pmatrix}5\\6\end{pmatrix}\)答案：B,D5.下列哪些矩陣是上三角矩陣？A.\(\begin{pmatrix}1&2\\3&4\end{pmatrix}\)B.\(\begin{pmatrix}1&0&0\\0&2&0\\0&0&3\end{pmatrix}\)C.\(\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}\)D.\(\begin{pmatrix}1&2\\0&3\end{pmatrix}\)答案：B,D6.下列哪些矩陣是下三角矩陣？A.\(\begin{pmatrix}1&2\\3&4\end{pmatrix}\)B.\(\begin{pmatrix}1&0&0\\0&2&0\\0&0&3\end{pmatrix}\)C.\(\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}\)D.\(\begin{pmatrix}1&0\\2&3\end{pmatrix}\)答案：D7.下列哪些矩陣是單位矩陣？A.\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\)B.\(\begin{pmatrix}2&0\\0&2\end{pmatrix}\)C.\(\begin{pmatrix}1&1\\1&1\end{pmatrix}\)D.\(\begin{pmatrix}0&1\\1&0\end{pmatrix}\)答案：A8.下列哪些矩陣是零矩陣？A.\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\)B.\(\begin{pmatrix}0&0\\0&0\end{pmatrix}\)C.\(\begin{pmatrix}2&3\\4&5\end{pmatrix}\)D.\(\begin{pmatrix}1&2\\3&4\end{pmatrix}\)答案：B9.下列哪些矩陣是正交矩陣？A.\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\)B.\(\begin{pmatrix}0&1\\1&0\end{pmatrix}\)C.\(\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\end{pmatrix}\)D.\(\begin{pmatrix}1&1\\1&-1\end{pmatrix}\)答案：A,B,C10.下列哪些矩陣是可逆的？A.\(\begin{pmatrix}1&2\\2&4\end{pmatrix}\)B.\(\begin{pmatrix}1&2\\3&4\end{pmatrix}\)C.\(\begin{pmatrix}0&1\\1&0\end{pmatrix}\)D.\(\begin{pmatrix}2&3\\4&6\end{pmatrix}\)答案：B,C三、判斷題（每題2分，共10題）1.任何方陣都有特征值。2.任何向量空間都有基。3.任何矩陣都可以對角化。4.任何線性變換都有特征值。5.任何多項式都可以分解為線性因子。6.任何向量空間都包含零向量。7.任何矩陣的秩等于其列向量組的秩。8.任何矩陣的行列式為零當(dāng)且僅當(dāng)其行列式為零。9.任何線性方程組都有解。10.任何向量空間都包含無窮多個向量。答案：1.錯，2.對，3.錯，4.錯，5.錯，6.對，7.對，8.對，9.錯，10.對四、簡答題（每題5分，共4題）1.簡述線性無關(guān)的定義。答案：向量組\(\{v_1,v_2,\ldots,v_n\}\)是線性無關(guān)的，如果對于任意不全為零的系數(shù)\(c_1,c_2,\ldots,c_n\)，都有\(zhòng)(c_1v_1+c_2v_2+\cdots+c_nv_n\n