高等代数题库及答案

高等代数题库及答案

一、单项选择题，(总共10题，每题2分)。1.在实数域上，下列哪个矩阵是可逆的？A.\(\begin{pmatrix}1&2\\3&4\end{pmatrix}\)B.\(\begin{pmatrix}0&1\\0&0\end{pmatrix}\)C.\(\begin{pmatrix}1&1\\1&1\end{pmatrix}\)D.\(\begin{pmatrix}2&3\\4&6\end{pmatrix}\)答案：A2.多项式\(p(x)=x^3-3x+2\)在实数域上的根是？A.1B.-1,1,2C.1,-2D.0,1,2答案：C3.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.\(\begin{pmatrix}0&1&2\\0&0&3\\0&0&0\end{pmatrix}\)答案：B5.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.\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\)D.\(\begin{pmatrix}1&-1\\-1&1\end{pmatrix}\)答案：C6.下列哪个矩阵是可对角化的？A.\(\begin{pmatrix}1&2\\0&1\end{pmatrix}\)B.\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\)C.\(\begin{pmatrix}1&1\\0&1\end{pmatrix}\)D.\(\begin{pmatrix}1&0\\0&2\end{pmatrix}\)答案：D7.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：C8.下列哪个向量是单位向量？A.\(\begin{pmatrix}1\\1\end{pmatrix}\)B.\(\begin{pmatrix}\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\end{pmatrix}\)C.\(\begin{pmatrix}2\\2\end{pmatrix}\)D.\(\begin{pmatrix}0\\1\end{pmatrix}\)答案：B9.下列哪个矩阵是幂等矩阵？A.\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\)B.\(\begin{pmatrix}0&1\\1&0\end{pmatrix}\)C.\(\begin{pmatrix}1&1\\1&0\end{pmatrix}\)D.\(\begin{pmatrix}1&0\\0&0\end{pmatrix}\)答案：D10.下列哪个矩阵是投影矩阵？A.\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\)B.\(\begin{pmatrix}0&1\\1&0\end{pmatrix}\)C.\(\begin{pmatrix}1&0\\0&0\end{pmatrix}\)D.\(\begin{pmatrix}1&1\\1&1\end{pmatrix}\)答案：C二、多项选择题，(总共10题，每题2分)。1.下列哪些矩阵是可逆的？A.\(\begin{pmatrix}1&2\\3&4\end{pmatrix}\)B.\(\begin{pmatrix}0&1\\0&0\end{pmatrix}\)C.\(\begin{pmatrix}1&1\\1&1\end{pmatrix}\)D.\(\begin{pmatrix}2&3\\4&6\end{pmatrix}\)答案：A2.多项式\(p(x)=x^2-4\)在实数域上的根是？A.2B.-2C.4D.-4答案：A,B3.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,C4.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.\(\begin{pmatrix}0&1&2\\0&0&3\\0&0&0\end{pmatrix}\)答案：B,D5.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.\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\)D.\(\begin{pmatrix}1&-1\\-1&1\end{pmatrix}\)答案：C6.下列哪些矩阵是可对角化的？A.\(\begin{pmatrix}1&2\\0&1\end{pmatrix}\)B.\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\)C.\(\begin{pmatrix}1&1\\0&1\end{pmatrix}\)D.\(\begin{pmatrix}1&0\\0&2\end{pmatrix}\)答案：B,D7.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,C8.下列哪些向量是单位向量？A.\(\begin{pmatrix}1\\1\end{pmatrix}\)B.\(\begin{pmatrix}\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\end{pmatrix}\)C.\(\begin{pmatrix}2\\2\end{pmatrix}\)D.\(\begin{pmatrix}0\\1\end{pmatrix}\)答案：B9.下列哪些矩阵是幂等矩阵？A.\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\)B.\(\begin{pmatrix}0&1\\1&0\end{pmatrix}\)C.\(\begin{pmatrix}1&1\\1&0\end{pmatrix}\)D.\(\begin{pmatrix}1&0\\0&0\end{pmatrix}\)答案：A,D10.下列哪些矩阵是投影矩阵？A.\(\begin{pmatrix}1&0\\0&1\end{pmatrix}\)B.\(\begin{pmatrix}0&1\\1&0\end{pmatrix}\)C.\(\begin{pmatrix}1&0\\0&0\end{pmatrix}\)D.\(\begin{pmatrix}1&1\\1&1\end{pmatrix}\)答案：C三、判断题，(总共10题，每题2分)。1.任何方阵都可以对角化。答案：错误2.如果一个矩阵的所有特征值都是正的，那么它是正定矩阵。答案：正确3.任何向量空间都至少有一个基。答案：正确4.如果一个向量组是线性无关的，那么它的任何子集也是线性无关的。答案：正确5.任何矩阵都可以相似对角化。答案：错误6.如果一个矩阵是正交矩阵，那么它的转置也是正交矩阵。答案：正确7.任何多项式都可以在实数域上分解为一次因式的乘积。答案：错误8.如果一个向量组是线性相关的，那么它的任何子集也是线性相关的。答案：正确9.任何矩阵的特征值都是实数。答案：错误10.如果一个矩阵是幂等矩阵，那么它的特征值只能是0或1。答案：正确四、简答题，(总共4题，每题5分)。1.简述矩阵的特征值和特征向量的定义。答案：矩阵的特征值和特征向量定义如下：对于方阵A，如果存在一个数λ和一个非零向量x，使得Ax=λx，那么λ称为A的特征值，x称为A对应于特征值λ的特征向量。2.简述向量空间的基本性质。答案：向量空间的基本性质包括：封闭性（加法和数乘的结果仍在向量空间中）、结合律（加法和数乘的结