2025年線性代數(shù)與空間解析幾何基礎理論測試試卷一、選擇題（每題3分，共30分）1.設向量$\boldsymbol{a}=(1,2,3)$，向量$\boldsymbol=(2,1,0)$，則$\boldsymbol{a}\cdot\boldsymbol$的值為：A.5B.7C.9D.112.設$\boldsymbol{A}$是一個$3\times3$的實對稱矩陣，且$\boldsymbol{A}^2=\boldsymbol{O}$，則$\boldsymbol{A}$的特征值是：A.0B.1C.-1D.$\pm1$3.設$\boldsymbol{A}$是一個$3\times3$的矩陣，且$\boldsymbol{A}^2=\boldsymbol{O}$，則$\boldsymbol{A}$的行列式$\left|\boldsymbol{A}\right|$的值為：A.0B.1C.-1D.無法確定4.設$\boldsymbol{A}$是一個$3\times3$的矩陣，且$\boldsymbol{A}^2=\boldsymbol{O}$，則$\boldsymbol{A}$的秩$r(\boldsymbol{A})$為：A.0B.1C.2D.35.設$\boldsymbol{A}$是一個$3\times3$的矩陣，且$\boldsymbol{A}^2=\boldsymbol{A}$，則$\boldsymbol{A}$的特征值是：A.0B.1C.-1D.$\pm1$6.設$\boldsymbol{A}$是一個$3\times3$的矩陣，且$\boldsymbol{A}^2=\boldsymbol{O}$，則$\boldsymbol{A}$的逆矩陣$\boldsymbol{A}^{-1}$存在的充要條件是：A.$\boldsymbol{A}$的行列式$\left|\boldsymbol{A}\right|\neq0$B.$\boldsymbol{A}$的行列式$\left|\boldsymbol{A}\right|=0$C.$\boldsymbol{A}$的秩$r(\boldsymbol{A})=3$D.$\boldsymbol{A}$的秩$r(\boldsymbol{A})\leq2$7.設$\boldsymbol{A}$是一個$3\times3$的矩陣，且$\boldsymbol{A}^2=\boldsymbol{O}$，則$\boldsymbol{A}$的伴隨矩陣$\boldsymbol{A}^*$的值是：A.$\boldsymbol{O}$B.$\boldsymbol{A}$C.$\boldsymbol{A}^2$D.$\boldsymbol{A}^3$8.設$\boldsymbol{A}$是一個$3\times3$的矩陣，且$\boldsymbol{A}^2=\boldsymbol{O}$，則$\boldsymbol{A}$的特征值$\lambda$滿足：A.$\lambda^2=0$B.$\lambda^2=1$C.$\lambda^2=-1$D.$\lambda^2=\pm1$9.設$\boldsymbol{A}$是一個$3\times3$的矩陣，且$\boldsymbol{A}^2=\boldsymbol{O}$，則$\boldsymbol{A}$的行列式$\left|\boldsymbol{A}\right|$的值為：A.0B.1C.-1D.無法確定10.設$\boldsymbol{A}$是一個$3\times3$的矩陣，且$\boldsymbol{A}^2=\boldsymbol{O}$，則$\boldsymbol{A}$的秩$r(\boldsymbol{A})$為：A.0B.1C.2D.3二、填空題（每題3分，共30分）1.設$\boldsymbol{a}=(1,2,3)$，$\boldsymbol=(2,1,0)$，則$\boldsymbol{a}\cdot\boldsymbol=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\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trix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{matrix}\right)=\left(\begin{matrix}a_{11}^2+a_{12}^2+a_{13}^2&a_{11}a_{21}+a_{12}a_{22}+a_{13}a_{23}&a_{11}a_{31}+a_{12}a_{32}+a_{13}a_{33}\\a_{21}a_{11}+a_{22}a_{12}+a_{23}a_{13}&a_{21}^2+a_{22}^2+a_{23}^2&a_{21}a_{31}+a_{22}a_{32}+a_{23}a_{33}\\a_{31}a_{11}+a_{32}a_{12}+a_{33}a_{13}&a_{31}a_{21}+a_{32}a_{22}+a_{33}a_{23}&a_{31}^2+a_{32}^2+a_{33}^2\end{matrix}\right)=\left(\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\right)=\boldsymbol{I}$。2.解析思路：利用實對稱矩陣的性質和特征值的定義進行證明。答案：設$\lambda$是$\boldsymbol{A}$的一個特征值，$\bold