《高等代数》题库及答案

《高等代数》题库及答案

一、单项选择题（每题2分，共10题）1.在实数域上，下列哪个矩阵是可逆的？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.多项式\(f(x)=x^3-3x+2\)在实数域上的根的个数是：A.0B.1C.2D.3答案：D3.下列哪个向量是线性无关的？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.下列哪个矩阵是正定矩阵？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.下列哪个向量是线性组合？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.下列哪个矩阵是上三角矩阵？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.下列哪个矩阵是下三角矩阵？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.下列哪个矩阵是单位矩阵？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.下列哪个矩阵是零矩阵？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.下列哪个矩阵是正交矩阵？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、多项选择题（每题2分，共10题）1.下列哪些矩阵是可逆的？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.下列哪些向量是线性无关的？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.下列哪些矩阵是正定矩阵？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.下列哪些向量是线性组合？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.下列哪些矩阵是上三角矩阵？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.下列哪些矩阵是下三角矩阵？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.下列哪些矩阵是单位矩阵？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.下列哪些矩阵是零矩阵？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.下列哪些矩阵是正交矩阵？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.下列哪些矩阵是可逆的？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、判断题（每题2分，共10题）1.任何方阵都有特征值。2.任何向量空间都有基。3.任何矩阵都可以对角化。4.任何线性变换都有特征值。5.任何多项式都可以分解为线性因子。6.任何向量空间都包含零向量。7.任何矩阵的秩等于其列向量组的秩。8.任何矩阵的行列式为零当且仅当其行列式为零。9.任何线性方程组都有解。10.任何向量空间都包含无穷多个向量。答案：1.错，2.对，3.错，4.错，5.错，6.对，7.对，8.对，9.错，10.对四、简答题（每题5分，共4题）1.简述线性无关的定义。答案：向量组\(\{v_1,v_2,\ldots,v_n\}\)是线性无关的，如果对于任意不全为零的系数\(c_1,c_2,\ldots,c_n\)，都有\(c_1v_1+c_2v_2+\cdots+c_nv_n\n