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문제

세 변수 \(x_1\), \(x_2\), \(x_3\)로 이루어진 일차 방정식 세 개가 있다. \(a_{ij}\)와 \(b_i\)가 주어졌을 때, 다음과 같이 표현할 수 있다.

\(\begin{vmatrix} a_{ 11 }x_{ 1 }+a_{ 12 }x_{ 2 }+a_{ 13 }x_{ 3 }=b_{ 1 } \\ a_{ 21 }x_{ 1 }+a_{ 22 }x_{ 2 }+a_{ 23 }x_{ 3 }=b_{ 2 } \\ a_{ 31 }x_{ 1 }+a_{ 32 }x_{ 2 }+a_{ 33 }x_{ 3 }=b_{ 3 } \end{vmatrix}\)

행렬과 벡터를 이용하면 다음과 같이 나타낼 수 있다.

\(A\cdot x=b\), where \(A=\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}\), \(b=\begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}\), x=\(\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\)

크래머의 공식을 이용하면 방정식의 해는 행렬식(determinant)를 이용해 다음과 같이 구할 수 있다.

\(x_i = \frac {det~A_i}{det~A}\)

\(A_i\) 는 행렬 \(A\)의 \(i\)번째 열을 벡터 \(b\)로 대체한 것이다. 예를 들어, \(3 \times 3\) 행렬의 행렬식은 아래와 같이 구할 수 있다.

\(det\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} = a_{11}(a_{22}a_{33} - a_{23}a_{32} - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32}-a_{22}a_{31})\)

크래머의 공식은 \(det~A \ne 0\)인 경우에만 해를 구할 수 있다. \(det~A \ne 0\)인 경우에 방정식의 해는 유일하고, 그렇지 않은 경우에는 해가 없거나 무수히 많은 해를 가진다.

방정식이 많은 경우에는 행렬식을 구하는데 시간이 너무 많이 걸린다. 따라서, 이런 경우에는 다른 방법을 사용하는 것이 더 효율적이다.

세 변수로 이루어진 일차 방정식 세 개가 주어진다. 크래머의 공식을 이용해서 해를 구하는 프로그램을 작성하시오.

입력

입력은 여러 개의 테스트 케이스로 이루어져 있다.

각 테스트 케이스는 세 줄로 이루어져 있고, 아래와 같은 순서로 주어진다.

\(\begin{matrix} a_{11} & a_{12} & a_{13} & b_1 \\ a_{21} & a_{22}  & a_{23} & b_{2}  \\ a_{31}  & a_{32} & a_{33} & b_{3}  \end{matrix}\)

모든 숫자는 -1000보다 크거나 같고, 1000보다 작거나 같은 정수이며, 공백 한 칸으로 구분되어져 있다. 

출력

각 테스트 케이스마다 두 줄을 출력한다.

첫째 줄에는 행렬 \(A_1\), \(A_2\), \(A_3\), \(A\)의 행렬식을 출력한다. 

방정식의 해가 존재하지 않는 경우에는 둘째 줄에 "No unique solution"을 출력하고, 해가 존재하는 경우에는 "Unique solution: "와 \(x_1\), \(x_2\), \(x_3\) 값을 소수점 셋째자리까지 출력한다.

방정식의 해 \(x_i\)가 -0.0005 < \(x_i\) < 0.0005 인 경우에는 "-0.000" 대신에 "0.000"을 출력한다.

각 테스트 케이스 사이에는 빈 줄을 하나 출력한다.

예제 입력 1

3
4 0 0 1
0 2 0 2
0 0 1 4
1 2 3 1
1 1 1 2
2 2 2 3
1 0 0 1
0 1 0 0
0 0 -1 0

예제 출력 1

2 8 32 8
Unique solution: 0.250 1.000 4.000

1 -2 1 0
No unique solution

-1 0 0 -1
Unique solution: 1.000 0.000 0.000
[{"problem_id":"7561","problem_lang":"0","title":"\ud06c\ub798\uba38\uc758 \uacf5\uc2dd","description":"<p>\uc138 \ubcc0\uc218 \\(x_1\\), \\(x_2\\), \\(x_3\\)\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc77c\ucc28 \ubc29\uc815\uc2dd \uc138 \uac1c\uac00 \uc788\ub2e4. \\(a_{ij}\\)\uc640 \\(b_i\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\ud604\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>\\(\\begin{vmatrix} a_{ 11 }x_{ 1 }+a_{ 12 }x_{ 2 }+a_{ 13 }x_{ 3 }=b_{ 1 } \\\\ a_{ 21 }x_{ 1 }+a_{ 22 }x_{ 2 }+a_{ 23 }x_{ 3 }=b_{ 2 } \\\\ a_{ 31 }x_{ 1 }+a_{ 32 }x_{ 2 }+a_{ 33 }x_{ 3 }=b_{ 3 } \\end{vmatrix}\\)<\/p>\r\n\r\n<p>\ud589\ub82c\uacfc \ubca1\ud130\ub97c \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>\\(A\\cdot x=b\\), where \\(A=\\begin{pmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\\\ a_{21} &amp; a_{22} &amp; a_{23} \\\\ a_{31} &amp; a_{32} &amp; a_{33} \\end{pmatrix}\\), \\(b=\\begin{pmatrix} b_1 \\\\ b_2 \\\\ b_3 \\end{pmatrix}\\), x=\\(\\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix}\\)<\/p>\r\n\r\n<p>\ud06c\ub798\uba38\uc758 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud558\uba74 \ubc29\uc815\uc2dd\uc758 \ud574\ub294 \ud589\ub82c\uc2dd(determinant)\ub97c \uc774\uc6a9\ud574 \ub2e4\uc74c\uacfc \uac19\uc774 \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>\\(x_i = \\frac {det~A_i}{det~A}\\)<\/p>\r\n\r\n<p>\\(A_i\\) \ub294 \ud589\ub82c \\(A\\)\uc758 \\(i\\)\ubc88\uc9f8 \uc5f4\uc744 \ubca1\ud130 \\(b\\)\ub85c \ub300\uccb4\ud55c \uac83\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \\(3 \\times 3\\) \ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc740 \uc544\ub798\uc640 \uac19\uc774 \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>\\(det\\begin{pmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\\\ a_{21} &amp; a_{22} &amp; a_{23} \\\\ a_{31} &amp; a_{32} &amp; a_{33} \\end{pmatrix} = a_{11}(a_{22}a_{33} - a_{23}a_{32} - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32}-a_{22}a_{31})\\)<\/p>\r\n\r\n<p>\ud06c\ub798\uba38\uc758 \uacf5\uc2dd\uc740 \\(det~A \\ne 0\\)\uc778 \uacbd\uc6b0\uc5d0\ub9cc \ud574\ub97c \uad6c\ud560 \uc218 \uc788\ub2e4. \\(det~A \\ne 0\\)\uc778 \uacbd\uc6b0\uc5d0 \ubc29\uc815\uc2dd\uc758 \ud574\ub294 \uc720\uc77c\ud558\uace0, \uadf8\ub807\uc9c0 \uc54a\uc740 \uacbd\uc6b0\uc5d0\ub294 \ud574\uac00 \uc5c6\uac70\ub098 \ubb34\uc218\ud788 \ub9ce\uc740 \ud574\ub97c \uac00\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\ubc29\uc815\uc2dd\uc774 \ub9ce\uc740 \uacbd\uc6b0\uc5d0\ub294 \ud589\ub82c\uc2dd\uc744 \uad6c\ud558\ub294\ub370 \uc2dc\uac04\uc774 \ub108\ubb34 \ub9ce\uc774 \uac78\ub9b0\ub2e4. \ub530\ub77c\uc11c, \uc774\ub7f0 \uacbd\uc6b0\uc5d0\ub294 \ub2e4\ub978 \ubc29\ubc95\uc744 \uc0ac\uc6a9\ud558\ub294 \uac83\uc774 \ub354 \ud6a8\uc728\uc801\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc138 \ubcc0\uc218\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc77c\ucc28 \ubc29\uc815\uc2dd \uc138 \uac1c\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \ud06c\ub798\uba38\uc758 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud574\uc11c \ud574\ub97c \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \uc5ec\ub7ec \uac1c\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \uc138 \uc904\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uace0, \uc544\ub798\uc640 \uac19\uc740 \uc21c\uc11c\ub85c \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\\(\\begin{matrix} a_{11} &amp; a_{12} &amp; a_{13} &amp; b_1 \\\\ a_{21} &amp; a_{22} &nbsp;&amp; a_{23} &amp; b_{2} &nbsp;\\\\ a_{31} &nbsp;&amp; a_{32} &amp; a_{33} &amp; b_{3} &nbsp;\\end{matrix}\\)<\/p>\r\n\r\n<p>\ubaa8\ub4e0 \uc22b\uc790\ub294 -1000\ubcf4\ub2e4 \ud06c\uac70\ub098 \uac19\uace0, 1000\ubcf4\ub2e4 \uc791\uac70\ub098 \uac19\uc740 \uc815\uc218\uc774\uba70, \uacf5\ubc31 \ud55c \uce78\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4\uc838 \uc788\ub2e4.&nbsp;<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 \ub450 \uc904\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uccab\uc9f8 \uc904\uc5d0\ub294 \ud589\ub82c \\(A_1\\), \\(A_2\\), \\(A_3\\), \\(A\\)\uc758 \ud589\ub82c\uc2dd\uc744 \ucd9c\ub825\ud55c\ub2e4.&nbsp;<\/p>\r\n\r\n<p>\ubc29\uc815\uc2dd\uc758 \ud574\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294 \uacbd\uc6b0\uc5d0\ub294 \ub458\uc9f8 \uc904\uc5d0 &quot;No unique solution&quot;\uc744 \ucd9c\ub825\ud558\uace0, \ud574\uac00 \uc874\uc7ac\ud558\ub294 \uacbd\uc6b0\uc5d0\ub294 &quot;Unique solution: &quot;\uc640 \\(x_1\\), \\(x_2\\), \\(x_3\\) \uac12\uc744 \uc18c\uc218\uc810 \uc14b\uc9f8\uc790\ub9ac\uae4c\uc9c0 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\ubc29\uc815\uc2dd\uc758 \ud574 \\(x_i\\)\uac00 -0.0005 &lt; \\(x_i\\) &lt; 0.0005 \uc778 \uacbd\uc6b0\uc5d0\ub294 &quot;-0.000&quot; \ub300\uc2e0\uc5d0 &quot;0.000&quot;\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4 \uc0ac\uc774\uc5d0\ub294 \ube48 \uc904\uc744 \ud558\ub098 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"7561","problem_lang":"1","title":"Cramer\u2019s Rule","description":"<p>Consider a system of linear equations, here three equations of three variables \\(x_1\\), \\(x_2\\), \\(x_3\\). The general form looks something like this, with given numbers \\(a_{ij}\\) and \\(b_i\\)<\/p>\r\n\r\n<p>\\(\\begin{vmatrix} a_{ 11 }x_{ 1 }+a_{ 12 }x_{ 2 }+a_{ 13 }x_{ 3 }=b_{ 1 } \\\\ a_{ 21 }x_{ 1 }+a_{ 22 }x_{ 2 }+a_{ 23 }x_{ 3 }=b_{ 2 } \\\\ a_{ 31 }x_{ 1 }+a_{ 32 }x_{ 2 }+a_{ 33 }x_{ 3 }=b_{ 3 } \\end{vmatrix}\\)<\/p>\r\n\r\n<p>Or, using matrices and vectors:<\/p>\r\n\r\n<p>\\(A\\cdot x=b\\), where \\(A=\\begin{pmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\\\ a_{21} &amp; a_{22} &amp; a_{23} \\\\ a_{31} &amp; a_{32} &amp; a_{33} \\end{pmatrix}\\), \\(b=\\begin{pmatrix} b_1 \\\\ b_2 \\\\ b_3 \\end{pmatrix}\\), x=\\(\\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix}\\)<\/p>\r\n\r\n<p>According to Cramer&rsquo;s rule, the solution can be given in terms of determinants, i.e.<\/p>\r\n\r\n<p>\\(x_i = \\frac {det~A_i}{det~A}\\)<\/p>\r\n\r\n<p>where \\(A_i\\) is the matrix obtained from \\(A\\) by replacing the \\(i\\)-th column with the vector \\(b\\). For \\(3 \\times 3\\) determinants, you can use the following simple formular to calculate the determinant:<\/p>\r\n\r\n<p>\\(det\\begin{pmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\\\ a_{21} &amp; a_{22} &amp; a_{23} \\\\ a_{31} &amp; a_{32} &amp; a_{33} \\end{pmatrix} = a_{11}(a_{22}a_{33} - a_{23}a_{32} - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32}-a_{22}a_{31})\\)<\/p>\r\n\r\n<p>Obviously, Cramer&rsquo;s rule only works for \\(det~A \\ne 0\\). One can show that the system has a unique solution if and only if \\(det~A \\ne 0\\). Otherwise, the system has either no solution or in\ufb01nitely many solutions.<\/p>\r\n\r\n<p>Please note that one would not use Cramer&rsquo;s rule to solve a large system of linear equations, simply because calculating a single determinant is as time-consuming as solving the complete system by a more ef\ufb01cient algorithm.<\/p>\r\n\r\n<p>Given a system of three linear equations in three variables, use Cramer&rsquo;s rule to \ufb01nd the unique solution if it exists. More precisely, calculate the determinants of the \\(A_i\\) and of \\(A\\) and decide by looking at \\(det~A\\) whether the system has a unique solution. If it does, calculate the solution according to Cramer&rsquo;s rule.<\/p>\r\n","input":"<p>The \ufb01rst line contains the number of scenarios.<\/p>\r\n\r\n<p>For each scenario, you are given three lines corresponding to the three equations, with the coef\ufb01cients of the matrix \\(A\\) and the coordinates of the vector \\(b\\) arranged as follows:<\/p>\r\n\r\n<p>\\(\\begin{matrix} a_{11} &amp; a_{12} &amp; a_{13} &amp; b_1 \\\\ a_{21} &amp; a_{22} &nbsp;&amp; a_{23} &amp; b_{2} &nbsp;\\\\ a_{31} &nbsp;&amp; a_{32} &amp; a_{33} &amp; b_{3} &nbsp;\\end{matrix}\\)<\/p>\r\n\r\n<p>All numbers are integers in the range {&minus;1000, ... ,1000}. They are separated by single blanks.<\/p>\r\n","output":"<p>For each scenario print three lines. In the \ufb01rst line, print the determinants of \\(A_1\\), \\(A_2\\), \\(A_3\\), and \\(A\\), as integers and separated by single blanks. In the second line, print (depending on \\(det~A\\)) either &ldquo;No unique solution&rdquo; or &ldquo;Unique solution: &rdquo;, followed by the values of \\(x_1\\), \\(x_2\\), \\(x_3\\) with three digits after the decimal point, again separated from each other by a single blank. For solutions \\(x_i\\) with &minus;0.0005 &lt; \\(x_i\\) &lt; 0.0005 always print &ldquo;0.000&rdquo; instead of the &ldquo;-0.000&rdquo; that your print command might come up with. The third line is empty.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]

출처

University > Tu-Darmstadt Programming Contest > TUD Contest 2001 2번

  • 문제를 번역한 사람: baekjoon
  • 잘못된 데이터를 찾은 사람: cubelover