{"result":"<br><a href='https:\/\/en.wikipedia.org\/wiki\/Bareiss_algorithm' target='_blank'><h5>Montante's method<\/a><\/h5><hr><math  xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'> <mi>  <\/mi> <mrow> <mo> ( <\/mo> <mtable columnalign=left><mtr><mtd> <mn style=\"border: 1px solid gray;padding:5px;\"> 2 <\/mn> <\/mtd><mtd> <mn style=\"\"> 1 <\/mn> <\/mtd><mtd> <mn style=\"\"> 3 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"\"> 1 <\/mn> <\/mtd><mtd> <mn style=\"\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"\"> 2 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"\"> 4 <\/mn> <\/mtd><mtd> <mn style=\"\"> 1 <\/mn> <\/mtd><mtd> <mn style=\"\"> 8 <\/mn> <\/mtd><\/mtr><\/mtable><mo> )<\/mo><\/mrow><\/math><span title='A[2][1] = (2*1 - 1*2) \/ 1 = 0&#013;A[2][2] = (2*0 - 1*1) \/ 1 = -1&#013;A[2][3] = (2*2 - 1*3) \/ 1 = 1&#013;A[3][1] = (2*4 - 4*2) \/ 1 = 0&#013;A[3][2] = (2*1 - 4*1) \/ 1 = -2&#013;A[3][3] = (2*8 - 4*3) \/ 1 = 4&#013;'><math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' ><mstyle displaystyle='true' scriptlevel='0'><mtable columnalign=left> <mtr> <mrow> <mfrac displaystyle='true' style='font-size:20px;color:blue;'> <mrow > \r\n            <msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mi> 1 <\/mi><mi> 1 <\/mi> <\/mrow> <\/msub> \r\n            <mo> &#x00D7; <!-- multiplication sign --> <\/mo> \r\n            <msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> ij <\/mn> <\/mrow> <\/msub> \r\n            <mo> - <\/mo> \r\n            <msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> i,1 <\/mn> <\/mrow> <\/msub> \r\n            <mo> &#x00D7; <!-- multiplication sign --> <\/mo> \r\n            <msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> 1j <\/mn> <\/mrow> <\/msub> \r\n            <\/mrow> <mrow> <msub> <mrow> <mi> P <\/mi> <\/mrow> <mrow> <mn> 0 <\/mn> <\/mrow> <\/msub> <\/mrow> <\/mfrac>  \r\n            <mo displaystyle='true' style='color:blue;'> &#x2192; <!-- rightwards arrow --> <\/mo> <msub displaystyle='true' style='color:blue;'> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> ij <\/mn> <\/mrow> <\/msub>\r\n            <\/mrow> <\/mtr> \r\n            \r\n            <mtr> \r\n            <mrow> <msub> <mrow> <mn>i<\/mn><mo>&#x2260; <!-- not equal to --> <\/mo><mn>1<\/mn> <mo>,<\/mo> <mi> P <\/mi> <\/mrow> <mrow> <mn> 1 <\/mn> <\/mrow> <\/msub> <mo> = <\/mo> <mn><msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> 11 <\/mn> <\/mrow> <\/msub> <\/mn> <mo>=<\/mo> <mn> 2 <\/mn> <\/mrow> <\/mtr> <\/mtable><\/mstyle>\r\n             <\/math> <\/math><\/span><math  xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'> <mi>  <\/mi> <mrow> <mo> ( <\/mo> <mtable columnalign=left><mtr><mtd> <mn style=\"\"> 2 <\/mn> <\/mtd><mtd> <mn style=\"\"> 1 <\/mn> <\/mtd><mtd> <mn style=\"\"> 3 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"color:blue;\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;border: 1px solid gray;padding:5px;\"> -1 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> 1 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"color:blue;\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> -2 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> 4 <\/mn> <\/mtd><\/mtr><\/mtable><mo> )<\/mo><\/mrow><\/math><span title='A[1][1] = (-1*2 - 1*0) \/ 2 = -1&#013;A[1][2] = (-1*1 - 1*-1) \/ 2 = 0&#013;A[1][3] = (-1*3 - 1*1) \/ 2 = -2&#013;A[3][1] = (-1*0 - -2*0) \/ 2 = 0&#013;A[3][2] = (-1*-2 - -2*-1) \/ 2 = 0&#013;A[3][3] = (-1*4 - -2*1) \/ 2 = -1&#013;'><math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' ><mstyle displaystyle='true' scriptlevel='0'><mtable columnalign=left> <mtr> <mrow> <mfrac displaystyle='true' style='font-size:20px;color:blue;'> <mrow > \r\n            <msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mi> 2 <\/mi><mi> 2 <\/mi> <\/mrow> <\/msub> \r\n            <mo> &#x00D7; <!-- multiplication sign --> <\/mo> \r\n            <msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> ij <\/mn> <\/mrow> <\/msub> \r\n            <mo> - <\/mo> \r\n            <msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> i,2 <\/mn> <\/mrow> <\/msub> \r\n            <mo> &#x00D7; <!-- multiplication sign --> <\/mo> \r\n            <msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> 2j <\/mn> <\/mrow> <\/msub> \r\n            <\/mrow> <mrow> <msub> <mrow> <mi> P <\/mi> <\/mrow> <mrow> <mn> 1 <\/mn> <\/mrow> <\/msub> <\/mrow> <\/mfrac>  \r\n            <mo displaystyle='true' style='color:blue;'> &#x2192; <!-- rightwards arrow --> <\/mo> <msub displaystyle='true' style='color:blue;'> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> ij <\/mn> <\/mrow> <\/msub>\r\n            <\/mrow> <\/mtr> \r\n            \r\n            <mtr> \r\n            <mrow> <msub> <mrow> <mn>i<\/mn><mo>&#x2260; <!-- not equal to --> <\/mo><mn>2<\/mn> <mo>,<\/mo> <mi> P <\/mi> <\/mrow> <mrow> <mn> 2 <\/mn> <\/mrow> <\/msub> <mo> = <\/mo> <mn><msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> 22 <\/mn> <\/mrow> <\/msub> <\/mn> <mo>=<\/mo> <mn> -1 <\/mn> <\/mrow> <\/mtr> <\/mtable><\/mstyle>\r\n             <\/math> <\/math><\/span><math  xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'> <mi>  <\/mi> <mrow> <mo> ( <\/mo> <mtable columnalign=left><mtr><mtd> <mn style=\"color:blue;\"> -1 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> -2 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"\"> -1 <\/mn> <\/mtd><mtd> <mn style=\"\"> 1 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"color:blue;\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;border: 1px solid gray;padding:5px;\"> -1 <\/mn> <\/mtd><\/mtr><\/mtable><mo> )<\/mo><\/mrow><\/math><span title='A[1][1] = (-1*-1 - -2*0) \/ -1 = -1&#013;A[1][2] = (-1*0 - -2*0) \/ -1 = 0&#013;A[1][3] = (-1*-2 - -2*-1) \/ -1 = 0&#013;A[2][1] = (-1*0 - 1*0) \/ -1 = 0&#013;A[2][2] = (-1*-1 - 1*0) \/ -1 = -1&#013;A[2][3] = (-1*1 - 1*-1) \/ -1 = 0&#013;'><math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML' ><mstyle displaystyle='true' scriptlevel='0'><mtable columnalign=left> <mtr> <mrow> <mfrac displaystyle='true' style='font-size:20px;color:blue;'> <mrow > \r\n            <msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mi> 3 <\/mi><mi> 3 <\/mi> <\/mrow> <\/msub> \r\n            <mo> &#x00D7; <!-- multiplication sign --> <\/mo> \r\n            <msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> ij <\/mn> <\/mrow> <\/msub> \r\n            <mo> - <\/mo> \r\n            <msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> i,3 <\/mn> <\/mrow> <\/msub> \r\n            <mo> &#x00D7; <!-- multiplication sign --> <\/mo> \r\n            <msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> 3j <\/mn> <\/mrow> <\/msub> \r\n            <\/mrow> <mrow> <msub> <mrow> <mi> P <\/mi> <\/mrow> <mrow> <mn> 2 <\/mn> <\/mrow> <\/msub> <\/mrow> <\/mfrac>  \r\n            <mo displaystyle='true' style='color:blue;'> &#x2192; <!-- rightwards arrow --> <\/mo> <msub displaystyle='true' style='color:blue;'> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> ij <\/mn> <\/mrow> <\/msub>\r\n            <\/mrow> <\/mtr> \r\n            \r\n            <mtr> \r\n            <mrow> <msub> <mrow> <mn>i<\/mn><mo>&#x2260; <!-- not equal to --> <\/mo><mn>3<\/mn> <mo>,<\/mo> <mi> P <\/mi> <\/mrow> <mrow> <mn> 3 <\/mn> <\/mrow> <\/msub> <mo> = <\/mo> <mn><msub> <mrow> <mi> a <\/mi> <\/mrow> <mrow> <mn> 33 <\/mn> <\/mrow> <\/msub> <\/mn> <mo>=<\/mo> <mn> -1 <\/mn> <\/mrow> <\/mtr> <\/mtable><\/mstyle>\r\n             <\/math> <\/math><\/span><math  xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'> <mi>  <\/mi> <mrow> <mo> ( <\/mo> <mtable columnalign=left><mtr><mtd> <mn style=\"color:blue;\"> -1 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> 0 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"color:blue;\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> -1 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> 0 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"\"> -1 <\/mn> <\/mtd><\/mtr><\/mtable><mo> )<\/mo><\/mrow><\/math><br><br><math  xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'> <mi> rank <\/mi> <mrow> <mo> ( <\/mo> <mtable columnalign=left><mtr><mtd> <mn> 2 <\/mn> <\/mtd><mtd> <mn> 1 <\/mn> <\/mtd><mtd> <mn> 3 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn> 1 <\/mn> <\/mtd><mtd> <mn> 0 <\/mn> <\/mtd><mtd> <mn> 2 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn> 4 <\/mn> <\/mtd><mtd> <mn> 1 <\/mn> <\/mtd><mtd> <mn> 8 <\/mn> <\/mtd><\/mtr><\/mtable><mo> )<\/mo><\/mrow><mo style='color:#111;'> = <\/mo><\/math> &nbsp;<math  xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'> <mi> rank <\/mi> <mrow> <mo> ( <\/mo> <mtable columnalign=left><mtr><mtd> <mn> -1 <\/mn> <\/mtd><mtd> <mn> 0 <\/mn> <\/mtd><mtd> <mn> 0 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn> 0 <\/mn> <\/mtd><mtd> <mn> -1 <\/mn> <\/mtd><mtd> <mn> 0 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn> 0 <\/mn> <\/mtd><mtd> <mn> 0 <\/mn> <\/mtd><mtd> <mn> -1 <\/mn> <\/mtd><\/mtr><\/mtable><mo> )<\/mo><\/mrow><mo style='color:#111;'> = <\/mo><\/math><math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'> <mn>3<\/mn> <\/math> <br><hr><a href='https:\/\/en.wikipedia.org\/wiki\/Gaussian_elimination' target='_blank'><h5>Gauss Elimination method<\/a><\/h5><hr><math  xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'> <mi>  <\/mi> <mrow> <mo> ( <\/mo> <mtable columnalign=left><mtr><mtd> <mn style=\"border: 1px solid gray;padding:5px;\"> 2 <\/mn> <\/mtd><mtd> <mn style=\"\"> 1 <\/mn> <\/mtd><mtd> <mn style=\"\"> 3 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"\"> 1 <\/mn> <\/mtd><mtd> <mn style=\"\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"\"> 2 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"\"> 4 <\/mn> <\/mtd><mtd> <mn style=\"\"> 1 <\/mn> <\/mtd><mtd> <mn style=\"\"> 8 <\/mn> <\/mtd><\/mtr><\/mtable><mo> )<\/mo><\/mrow><\/math><span title='Substract 2 row 1 from row 2&#013;A[2][1] = 1- (1*2) \/ 2 = 0&#013;A[2][2] = 0- (1*1) \/ 2 = -1\/2&#013;A[2][3] = 2- (1*3) \/ 2 = 1\/2&#013;'><math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'><mn ><\/mn><mn >R2<\/mn><mo>-<\/mo><mn>(1\/2)<\/mn><mn>R1<\/mn>\r\n                <mo>-><\/mo> <mn>R2<\/mn><\/math><\/span><math  xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'> <mi>  <\/mi> <mrow> <mo> ( <\/mo> <mtable columnalign=left><mtr><mtd> <mn style=\"border: 1px solid gray;padding:5px;\"> 2 <\/mn> <\/mtd><mtd> <mn style=\"\"> 1 <\/mn> <\/mtd><mtd> <mn style=\"\"> 3 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"color:blue;\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> -1\/2 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> 1\/2 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"\"> 4 <\/mn> <\/mtd><mtd> <mn style=\"\"> 1 <\/mn> <\/mtd><mtd> <mn style=\"\"> 8 <\/mn> <\/mtd><\/mtr><\/mtable><mo> )<\/mo><\/mrow><\/math><span title='Substract 2 row 1 from row 3&#013;A[3][1] = 4- (4*2) \/ 2 = 0&#013;A[3][2] = 1- (4*1) \/ 2 = -1&#013;A[3][3] = 8- (4*3) \/ 2 = 2&#013;'><math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'><mn ><\/mn><mn >R3<\/mn><mo>-<\/mo><mn>(2)<\/mn><mn>R1<\/mn>\r\n                <mo>-><\/mo> <mn>R3<\/mn><\/math><\/span><math  xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'> <mi>  <\/mi> <mrow> <mo> ( <\/mo> <mtable columnalign=left><mtr><mtd> <mn style=\"\"> 2 <\/mn> <\/mtd><mtd> <mn style=\"\"> 1 <\/mn> <\/mtd><mtd> <mn style=\"\"> 3 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"border: 1px solid gray;padding:5px;\"> -1\/2 <\/mn> <\/mtd><mtd> <mn style=\"\"> 1\/2 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"color:blue;\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> -1 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> 2 <\/mn> <\/mtd><\/mtr><\/mtable><mo> )<\/mo><\/mrow><\/math><span title='Substract -1\/2 row 2 from row 3&#013;A[3][2] = -1- (-1*-1\/2) \/ -1\/2 = 0&#013;A[3][3] = 2- (-1*1\/2) \/ -1\/2 = 1&#013;'><math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'><mn ><\/mn><mn >R3<\/mn><mo>-<\/mo><mn>(2)<\/mn><mn>R2<\/mn>\r\n                <mo>-><\/mo> <mn>R3<\/mn><\/math><\/span><math  xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'> <mi>  <\/mi> <mrow> <mo> ( <\/mo> <mtable columnalign=left><mtr><mtd> <mn style=\"\"> 2 <\/mn> <\/mtd><mtd> <mn style=\"\"> 1 <\/mn> <\/mtd><mtd> <mn style=\"\"> 3 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"\"> -1\/2 <\/mn> <\/mtd><mtd> <mn style=\"\"> 1\/2 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn style=\"color:blue;\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> 0 <\/mn> <\/mtd><mtd> <mn style=\"color:blue;\"> 1 <\/mn> <\/mtd><\/mtr><\/mtable><mo> )<\/mo><\/mrow><\/math><br><br><math  xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'> <mi> rank <\/mi> <mrow> <mo> ( <\/mo> <mtable columnalign=left><mtr><mtd> <mn> 2 <\/mn> <\/mtd><mtd> <mn> 1 <\/mn> <\/mtd><mtd> <mn> 3 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn> 1 <\/mn> <\/mtd><mtd> <mn> 0 <\/mn> <\/mtd><mtd> <mn> 2 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn> 4 <\/mn> <\/mtd><mtd> <mn> 1 <\/mn> <\/mtd><mtd> <mn> 8 <\/mn> <\/mtd><\/mtr><\/mtable><mo> )<\/mo><\/mrow><mo style='color:#111;'> = <\/mo><\/math> &nbsp;<math  xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'> <mi> rank <\/mi> <mrow> <mo> ( <\/mo> <mtable columnalign=left><mtr><mtd> <mn> 2 <\/mn> <\/mtd><mtd> <mn> 1 <\/mn> <\/mtd><mtd> <mn> 3 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn> 0 <\/mn> <\/mtd><mtd> <mn> -1\/2 <\/mn> <\/mtd><mtd> <mn> 1\/2 <\/mn> <\/mtd><\/mtr><mtr><mtd> <mn> 0 <\/mn> <\/mtd><mtd> <mn> 0 <\/mn> <\/mtd><mtd> <mn> 1 <\/mn> <\/mtd><\/mtr><\/mtable><mo> )<\/mo><\/mrow><mo style='color:#111;'> = <\/mo><\/math><math xmlns='http:\/\/www.w3.org\/1998\/Math\/MathML'> <mn>3<\/mn> <\/math> "}