求逆矩阵：
输入一个可逆矩阵，每个元用逗号隔开，每行用分号结尾。
注意，不支持支持数学函数和变量。
输入一个可逆矩阵，每个元用逗号隔开，每行用分号结尾。
注意，不支持支持数学函数和变量。
$$\begin{aligned}&\\ \color{black}{计算矩阵}& \ \ \begin{pmatrix} &17\ &24\ &1\ &8\ &15\ \\ &23\ &5\ &7\ &14\ &16\ \\ &4\ &6\ &13\ &20\ &22\ \\ &10\ &12\ &19\ &21\ &3\ \\ &11\ &18\ &25\ &2\ &9\ \end{pmatrix}\color{black}{的逆矩阵。}\\ \\解:&\\ &\begin{pmatrix} &17\ &24\ &1\ &8\ &15\ \\ &23\ &5\ &7\ &14\ &16\ \\ &4\ &6\ &13\ &20\ &22\ \\ &10\ &12\ &19\ &21\ &3\ \\ &11\ &18\ &25\ &2\ &9\ \end{pmatrix}\\\\&\color{grey}{用矩阵的初等变换来求逆矩阵：}\\&\left (\begin{array} {cccccc | ccccc} &17\ &24\ &1\ &8\ &15\ &1\ &0\ &0\ &0\ &0\ \\ &23\ &5\ &7\ &14\ &16\ &0\ &1\ &0\ &0\ &0\ \\ &4\ &6\ &13\ &20\ &22\ &0\ &0\ &1\ &0\ &0\ \\ &10\ &12\ &19\ &21\ &3\ &0\ &0\ &0\ &1\ &0\ \\ &11\ &18\ &25\ &2\ &9\ &0\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\&\color{grey}{将已知矩阵化为上三角矩阵}\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &17\ &24\ &1\ &8\ &15\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &-\frac{467}{17}\ &\frac{96}{17}\ &\frac{54}{17}\ &-\frac{73}{17}\ &-\frac{23}{17}\ &1\ &0\ &0\ &0\ \\ &0\ &\frac{6}{17}\ &\frac{217}{17}\ &\frac{308}{17}\ &\frac{314}{17}\ &-\frac{4}{17}\ &0\ &1\ &0\ &0\ \\ &0\ &-\frac{36}{17}\ &\frac{313}{17}\ &\frac{277}{17}\ &-\frac{99}{17}\ &-\frac{10}{17}\ &0\ &0\ &1\ &0\ \\ &0\ &\frac{42}{17}\ &\frac{414}{17}\ &-\frac{54}{17}\ &-\frac{12}{17}\ &-\frac{11}{17}\ &0\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &17\ &24\ &1\ &8\ &15\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &-\frac{467}{17}\ &\frac{96}{17}\ &\frac{54}{17}\ &-\frac{73}{17}\ &-\frac{23}{17}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &\frac{5995}{467}\ &\frac{8480}{467}\ &\frac{8600}{467}\ &-\frac{118}{467}\ &\frac{6}{467}\ &1\ &0\ &0\ \\ &0\ &0\ &\frac{8395}{467}\ &\frac{7495}{467}\ &-\frac{2565}{467}\ &-\frac{226}{467}\ &-\frac{36}{467}\ &0\ &1\ &0\ \\ &0\ &0\ &\frac{11610}{467}\ &-\frac{1350}{467}\ &-\frac{510}{467}\ &-\frac{359}{467}\ &\frac{42}{467}\ &0\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &17\ &24\ &1\ &8\ &15\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &-\frac{467}{17}\ &\frac{96}{17}\ &\frac{54}{17}\ &-\frac{73}{17}\ &-\frac{23}{17}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &\frac{5995}{467}\ &\frac{8480}{467}\ &\frac{8600}{467}\ &-\frac{118}{467}\ &\frac{6}{467}\ &1\ &0\ &0\ \\ &0\ &0\ &0\ &-\frac{11245}{1199}\ &-\frac{37505}{1199}\ &-\frac{72852}{559933}\ &-\frac{53238}{559933}\ &-\frac{1679}{1199}\ &1\ &0\ \\ &0\ &0\ &0\ &-\frac{45630}{1199}\ &-\frac{44070}{1199}\ &-\frac{156445}{559933}\ &\frac{36426}{559933}\ &-\frac{2322}{1199}\ &0\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &17\ &24\ &1\ &8\ &15\ &1\ &0\ &0\ &0\ &0\ \\ &0\ &-\frac{467}{17}\ &\frac{96}{17}\ &\frac{54}{17}\ &-\frac{73}{17}\ &-\frac{23}{17}\ &1\ &0\ &0\ &0\ \\ &0\ &0\ &\frac{5995}{467}\ &\frac{8480}{467}\ &\frac{8600}{467}\ &-\frac{118}{467}\ &\frac{6}{467}\ &1\ &0\ &0\ \\ &0\ &0\ &0\ &-\frac{11245}{1199}\ &-\frac{37505}{1199}\ &-\frac{72852}{559933}\ &-\frac{53238}{559933}\ &-\frac{1679}{1199}\ &1\ &0\ \\ &0\ &0\ &0\ &0\ &\frac{15600}{173}\ &\frac{20081}{80791}\ &\frac{36426}{80791}\ &\frac{648}{173}\ &-\frac{327834}{80791}\ &1\ \\\end{array} \right )\\\\&\color{grey}{将对角线以上的元素化为0}\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &17\ &24\ &1\ &8\ &0\ &\frac{997}{1040}\ &-\frac{1401}{18680}\ &-\frac{14013}{22490}\ &\frac{12609}{18680}\ &-\frac{173}{1040}\ \\ &0\ &-\frac{467}{17}\ &\frac{96}{17}\ &\frac{54}{17}\ &0\ &-\frac{355661}{265200}\ &\frac{3473}{3400}\ &\frac{1971}{11050}\ &-\frac{657}{3400}\ &\frac{12629}{265200}\ \\ &0\ &0\ &\frac{5995}{467}\ &\frac{8480}{467}\ &0\ &-\frac{11053}{36426}\ &-\frac{17279}{218089}\ &\frac{1427}{6071}\ &\frac{180729}{218089}\ &-\frac{7439}{36426}\ \\ &0\ &0\ &0\ &-\frac{11245}{1199}\ &0\ &-\frac{12629}{287760}\ &\frac{2941}{47960}\ &-\frac{1211}{11990}\ &-\frac{19549}{47960}\ &\frac{99821}{287760}\ \\ &0\ &0\ &0\ &0\ &\frac{15600}{173}\ &\frac{20081}{80791}\ &\frac{36426}{80791}\ &\frac{648}{173}\ &-\frac{327834}{80791}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &17\ &24\ &1\ &0\ &0\ &\frac{14371}{15600}\ &-\frac{27553}{1214200}\ &-\frac{461}{650}\ &\frac{851}{2600}\ &\frac{2021}{15600}\ \\ &0\ &-\frac{467}{17}\ &\frac{96}{17}\ &0\ &0\ &-\frac{359603}{265200}\ &\frac{46067}{44200}\ &\frac{1593}{11050}\ &-\frac{14643}{44200}\ &\frac{43787}{265200}\ \\ &0\ &0\ &\frac{5995}{467}\ &0\ &0\ &-\frac{70741}{182130}\ &\frac{1199}{30355}\ &\frac{1199}{30355}\ &\frac{1199}{30355}\ &\frac{85129}{182130}\ \\ &0\ &0\ &0\ &-\frac{11245}{1199}\ &0\ &-\frac{12629}{287760}\ &\frac{2941}{47960}\ &-\frac{1211}{11990}\ &-\frac{19549}{47960}\ &\frac{99821}{287760}\ \\ &0\ &0\ &0\ &0\ &\frac{15600}{173}\ &\frac{20081}{80791}\ &\frac{36426}{80791}\ &\frac{648}{173}\ &-\frac{327834}{80791}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &17\ &24\ &0\ &0\ &0\ &\frac{14843}{15600}\ &-\frac{31289}{1214200}\ &-\frac{463}{650}\ &\frac{843}{2600}\ &\frac{1453}{15600}\ \\ &0\ &-\frac{467}{17}\ &0\ &0\ &0\ &-\frac{314291}{265200}\ &\frac{45299}{44200}\ &\frac{1401}{11050}\ &-\frac{15411}{44200}\ &-\frac{10741}{265200}\ \\ &0\ &0\ &\frac{5995}{467}\ &0\ &0\ &-\frac{70741}{182130}\ &\frac{1199}{30355}\ &\frac{1199}{30355}\ &\frac{1199}{30355}\ &\frac{85129}{182130}\ \\ &0\ &0\ &0\ &-\frac{11245}{1199}\ &0\ &-\frac{12629}{287760}\ &\frac{2941}{47960}\ &-\frac{1211}{11990}\ &-\frac{19549}{47960}\ &\frac{99821}{287760}\ \\ &0\ &0\ &0\ &0\ &\frac{15600}{173}\ &\frac{20081}{80791}\ &\frac{36426}{80791}\ &\frac{648}{173}\ &-\frac{327834}{80791}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &17\ &0\ &0\ &0\ &0\ &-\frac{1309}{15600}\ &\frac{2261}{2600}\ &-\frac{182597}{303550}\ &\frac{23817}{1214200}\ &\frac{901}{15600}\ \\ &0\ &-\frac{467}{17}\ &0\ &0\ &0\ &-\frac{314291}{265200}\ &\frac{45299}{44200}\ &\frac{1401}{11050}\ &-\frac{15411}{44200}\ &-\frac{10741}{265200}\ \\ &0\ &0\ &\frac{5995}{467}\ &0\ &0\ &-\frac{70741}{182130}\ &\frac{1199}{30355}\ &\frac{1199}{30355}\ &\frac{1199}{30355}\ &\frac{85129}{182130}\ \\ &0\ &0\ &0\ &-\frac{11245}{1199}\ &0\ &-\frac{12629}{287760}\ &\frac{2941}{47960}\ &-\frac{1211}{11990}\ &-\frac{19549}{47960}\ &\frac{99821}{287760}\ \\ &0\ &0\ &0\ &0\ &\frac{15600}{173}\ &\frac{20081}{80791}\ &\frac{36426}{80791}\ &\frac{648}{173}\ &-\frac{327834}{80791}\ &1\ \\\end{array} \right )\\\\&\color{grey}{将主对角线元素化为1}\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &-\frac{77}{15600}\ &\frac{133}{2600}\ &-\frac{10741}{303550}\ &\frac{1401}{1214200}\ &\frac{53}{15600}\ \\ &0\ &-\frac{467}{17}\ &0\ &0\ &0\ &-\frac{314291}{265200}\ &\frac{45299}{44200}\ &\frac{1401}{11050}\ &-\frac{15411}{44200}\ &-\frac{10741}{265200}\ \\ &0\ &0\ &\frac{5995}{467}\ &0\ &0\ &-\frac{70741}{182130}\ &\frac{1199}{30355}\ &\frac{1199}{30355}\ &\frac{1199}{30355}\ &\frac{85129}{182130}\ \\ &0\ &0\ &0\ &-\frac{11245}{1199}\ &0\ &-\frac{12629}{287760}\ &\frac{2941}{47960}\ &-\frac{1211}{11990}\ &-\frac{19549}{47960}\ &\frac{99821}{287760}\ \\ &0\ &0\ &0\ &0\ &\frac{15600}{173}\ &\frac{20081}{80791}\ &\frac{36426}{80791}\ &\frac{648}{173}\ &-\frac{327834}{80791}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &-\frac{77}{15600}\ &\frac{133}{2600}\ &-\frac{10741}{303550}\ &\frac{1401}{1214200}\ &\frac{53}{15600}\ \\ &0\ &1\ &0\ &0\ &0\ &\frac{673}{15600}\ &-\frac{45299}{1214200}\ &-\frac{1401}{303550}\ &\frac{15411}{1214200}\ &\frac{10741}{7285200}\ \\ &0\ &0\ &\frac{5995}{467}\ &0\ &0\ &-\frac{70741}{182130}\ &\frac{1199}{30355}\ &\frac{1199}{30355}\ &\frac{1199}{30355}\ &\frac{85129}{182130}\ \\ &0\ &0\ &0\ &-\frac{11245}{1199}\ &0\ &-\frac{12629}{287760}\ &\frac{2941}{47960}\ &-\frac{1211}{11990}\ &-\frac{19549}{47960}\ &\frac{99821}{287760}\ \\ &0\ &0\ &0\ &0\ &\frac{15600}{173}\ &\frac{20081}{80791}\ &\frac{36426}{80791}\ &\frac{648}{173}\ &-\frac{327834}{80791}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &-\frac{77}{15600}\ &\frac{133}{2600}\ &-\frac{10741}{303550}\ &\frac{1401}{1214200}\ &\frac{53}{15600}\ \\ &0\ &1\ &0\ &0\ &0\ &\frac{673}{15600}\ &-\frac{45299}{1214200}\ &-\frac{1401}{303550}\ &\frac{15411}{1214200}\ &\frac{10741}{7285200}\ \\ &0\ &0\ &1\ &0\ &0\ &-\frac{27553}{910650}\ &\frac{1}{325}\ &\frac{1}{325}\ &\frac{1}{325}\ &\frac{33157}{910650}\ \\ &0\ &0\ &0\ &-\frac{11245}{1199}\ &0\ &-\frac{12629}{287760}\ &\frac{2941}{47960}\ &-\frac{1211}{11990}\ &-\frac{19549}{47960}\ &\frac{99821}{287760}\ \\ &0\ &0\ &0\ &0\ &\frac{15600}{173}\ &\frac{20081}{80791}\ &\frac{36426}{80791}\ &\frac{648}{173}\ &-\frac{327834}{80791}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &-\frac{77}{15600}\ &\frac{133}{2600}\ &-\frac{10741}{303550}\ &\frac{1401}{1214200}\ &\frac{53}{15600}\ \\ &0\ &1\ &0\ &0\ &0\ &\frac{673}{15600}\ &-\frac{45299}{1214200}\ &-\frac{1401}{303550}\ &\frac{15411}{1214200}\ &\frac{10741}{7285200}\ \\ &0\ &0\ &1\ &0\ &0\ &-\frac{27553}{910650}\ &\frac{1}{325}\ &\frac{1}{325}\ &\frac{1}{325}\ &\frac{33157}{910650}\ \\ &0\ &0\ &0\ &1\ &0\ &\frac{12629}{2698800}\ &-\frac{2941}{449800}\ &\frac{1211}{112450}\ &\frac{19549}{449800}\ &-\frac{577}{15600}\ \\ &0\ &0\ &0\ &0\ &\frac{15600}{173}\ &\frac{20081}{80791}\ &\frac{36426}{80791}\ &\frac{648}{173}\ &-\frac{327834}{80791}\ &1\ \\\end{array} \right )\\\\->\ \ &\left (\begin{array} {cccccc | ccccc} &1\ &0\ &0\ &0\ &0\ &-\frac{77}{15600}\ &\frac{133}{2600}\ &-\frac{10741}{303550}\ &\frac{1401}{1214200}\ &\frac{53}{15600}\ \\ &0\ &1\ &0\ &0\ &0\ &\frac{673}{15600}\ &-\frac{45299}{1214200}\ &-\frac{1401}{303550}\ &\frac{15411}{1214200}\ &\frac{10741}{7285200}\ \\ &0\ &0\ &1\ &0\ &0\ &-\frac{27553}{910650}\ &\frac{1}{325}\ &\frac{1}{325}\ &\frac{1}{325}\ &\frac{33157}{910650}\ \\ &0\ &0\ &0\ &1\ &0\ &\frac{12629}{2698800}\ &-\frac{2941}{449800}\ &\frac{1211}{112450}\ &\frac{19549}{449800}\ &-\frac{577}{15600}\ \\ &0\ &0\ &0\ &0\ &1\ &\frac{20081}{7285200}\ &\frac{1}{200}\ &\frac{4671}{112450}\ &-\frac{4203}{93400}\ &\frac{173}{15600}\ \\\end{array} \right )\\\\&\color{grey}{所求的逆矩阵为：}\\&\begin{pmatrix} &-\frac{77}{15600}\ &\frac{133}{2600}\ &-\frac{10741}{303550}\ &\frac{1401}{1214200}\ &\frac{53}{15600}\ \\ &\frac{673}{15600}\ &-\frac{45299}{1214200}\ &-\frac{1401}{303550}\ &\frac{15411}{1214200}\ &\frac{10741}{7285200}\ \\ &-\frac{27553}{910650}\ &\frac{1}{325}\ &\frac{1}{325}\ &\frac{1}{325}\ &\frac{33157}{910650}\ \\ &\frac{12629}{2698800}\ &-\frac{2941}{449800}\ &\frac{1211}{112450}\ &\frac{19549}{449800}\ &-\frac{577}{15600}\ \\ &\frac{20081}{7285200}\ &\frac{1}{200}\ &\frac{4671}{112450}\ &-\frac{4203}{93400}\ &\frac{173}{15600}\ \end{pmatrix}\end{aligned}$$
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矩阵的初等变换：
定义：对矩阵的行（列）施行下列三种变换都成为矩阵的初等变换：
（1）互换矩阵两行（列）的位置；
（2）用非零常数λ乘矩阵的某行（列）；
（3）将矩阵某行（列）的γ倍加到矩阵的另一行（列）上。
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