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问题答案
线性方程组中的未知数个数和方程式个数是否相等?可以相等,也可以不相等。
在矩阵形式线性方程组Ax = b中,A代表什么?系数矩阵。
在矩阵形式线性方程组Ax = b中,x代表什么?未知数向量。
在矩阵形式线性方程组Ax = b中,b代表什么?常数向量。
线性方程组的解?唯一解,无解或无穷多个解。
n元线性方程组的标准式是什么样子的?$$ \begin{array}{c}a_{11} x_{1}+a_{12} x_{2}+\cdots+a_{1 n} x_{n}=b_{1} \ a_{21} x_{1}+a_{22} x_{2}+\cdots+a_{2 n} x_{n}=b_{2} \ \cdots \cdots \cdots\a_{m1} x_{1}+a_{m2} x_{2}+\cdots+a_{m n} x_{n}=b_{m}\end{array} $$
$$ \left|\begin{array}{ll}a & b \ c & d\end{array}\right|= $$$$ad-bc$$
设有二元一次方程组$$ \left{\begin{array}{l}a_{11} x_{1}+a_{12} x_{2}=b_{1} \ a_{21} x_{1}+a_{22} x_{2}=b_{2}\end{array}\right. $$求写出克莱姆形式的解并校对自己推导过程?用 $$ a_{22} $$乘第一式的两边,用 $$ -a_{12}$$乘第二式的两边得:$$\left{\begin{array}{l}a_{11} a_{22} x_{1}+a_{12} a_{22} x_{2}=b_{1} a_{22}, \-a_{12} a_{21} x_{1}-a_{12} a_{22} x_{2}=-b_{2} a_{12} .\end{array}\right.$$将这两个方程式两边相加得:$$\left(a_{11} a_{22}-a_{12} a_{21}\right) x_{1}=b_{1} a_{22}-b_{2} a_{12} .$$于是$$x_{1}=\frac{b_{1} a_{22}-b_{2} a_{12}}{a_{11} a_{22}-a_{12} a_{21}} .$$用类似的办法消去 $$ x_{1} $$解得:$$x_{2}=\frac{a_{11} b_{2}-a_{21} b_{1}}{a_{11} a_{22}-a_{12} a_{21}} .$$
如何记忆二元一次方程组:$$\left{\begin{array}{l}a_{11} x_{1}+a_{12} x_{2}=b_{1} \ a_{21} x_{1}+a_{22} x_{2}=b_{2}\end{array}\right.$$ $$x_{1}=\frac{\left|\begin{array}{ll}b_{1}&a_{12}\ b_{2} & a_{22}\end{array}\right|}{\left|\begin{array}{ll}a_{11}&a_{12}\ a_{21} & a_{22}\end{array}\right|}, x_{2}=\frac{\left|\begin{array}{ll}a_{11} & b_{1} \ a_{21} & b_{2}\end{array}\right|}{\left|\begin{array}{ll}a_{11} & a_{12} \ a_{21} & a_{22}\end{array}\right|} $$(1)x1 与x2 的分母都是行列式$$ \left|\begin{array}{ll}a_{11} & a_{12} \ a_{21} & a_{22}\end{array}\right| $$ ,即只需将原方程组未知数前的系数按原顺序排成一个行列式即可.(2)x1 的分子行列式的第一列是原方程组的常数列,第二列由x2 的系数组成,因此这个行列式可以看成是将x1 与x2 的分母行列式$$ \left|\begin{array}{ll}a_{11} & a_{12} \ a_{21} & a_{22}\end{array}\right| $$中的第一列换成常数项而得.这个规则对x2 的分子行列式也适用.
设有二阶行列式$$ |A|=\left|\begin{array}{cc}a_{11} & a_{12} \ 0 & a_{22}\end{array}\right| $$ $$a_{11}$$和 $$a_{22}$$被称为什么?对角线元素或主对角元素
上三角行列式的值等于……元素之积其对角线
行列式某行或某列全为零,则行列式的值等于?0
用常数c乘以行列式的某一行或某一列,得到的行列式的值与原来的行列式的关系?是原行列式的值的c倍.$$ |\begin{array}{cc}\mathrm{ca}{11} & \mathrm{ca}{12} \ \mathrm{a}{21} & \mathrm{a}{22}\end{array}\left|=\left(\mathrm{ca}{11}\right) \mathrm{a}{22}-\left(\mathrm{ca}{12}\right) \mathrm{a}{21}=\mathrm{c|A}\right| $$
交换行列式不同的两行(列),行列式的值如何变化?改变正负
行列式两行或两列成比例(相同则当为比例为1),则行列式的值等于?0
请问$$ \left|\begin{array}{ll}a_{11}+b_{11} & a_{12}+b_{12} \ a_{21}+b_{21} & a_{22}+b_{22}\end{array}\right|=\left|\begin{array}{ll}a_{11} & a_{12} \ a_{21} & a_{22}\end{array}\right|+\left|\begin{array}{ll}b_{11} & b_{12} \ b_{21} & b_{22}\end{array}\right| $$是否成立?如不成立,请写出对应正确的行列式性质不正确,正确形式是$$ \begin{array}{l}\left|\begin{array}{cc}\mathrm{a}{11} & a{12} \ b_{21}+c_{21} & b_{22}+c_{22}\end{array}\right|=\left|\begin{array}{ll}a_{11} & a_{12} \ b_{21} & b_{22}\end{array}\right|+\left|\begin{array}{cc}a_{11} & a_{12} \ c_{21} & c_{22}\end{array}\right| ; \ \left|\begin{array}{ll}b_{11}+c_{11} & a_{12} \ b_{21}+c_{21} & a_{22}\end{array}\right|=\left|\begin{array}{cc}b_{11} & a_{12} \ b_{21} & a_{22}\end{array}\right|+\left|\begin{array}{cc}c_{11} & a_{12} \ c_{21} & a_{22}\end{array}\right| .\end{array} $$
行列式的某一行(列)乘以某个数加到另一行(列)上,行列式的值如何变化?不变
若行列式中某行(列)元素均为两项之和,则行列式可表示为?两个行列式之和
设有二阶行列式$$ |A|=\left|\begin{array}{ll}a_{11} & a_{12} \ a_{21} & a_{22}\end{array}\right| $$,求$$ |A|$$的转置$$ \left|\begin{array}{ll}a_{11} & a_{21} \ a_{12} & a_{22}\end{array}\right| $$
行列式和其转置的值关系是?相同
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