(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "times"; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "times"; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, 12, "courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, "courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, "courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, "courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "courier"; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B32768, 10, "times"; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 12, "times"; fontset = leftheader, 12, "times"; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, 12, "times"; fontset = leftfooter, 12, "times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "courier"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "times"; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] Computer HW #1 due 9/16 :[font = input; preserveAspect] Your Name Here. ;[s] 2:0,0;17,1;19,-1; 2:1,19,14,Times,1,18,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] Due 9/16. Worth 5 extra credit points. :[font = input; preserveAspect] There are 5 systems of linear equations. For each one, find the matricies Mand b such that Mx= b is the system (where x is an column of the appropriate variables), show this is the case, find the augmented matrix AugA of the system, write AugA in reduced row-echelon form, find all solutions of the system, and if possible, compute M^2 and the inverse of M. Print out your solutions. ;[s] 19:0,0;75,1;76,0;81,1;82,0;95,1;98,0;99,1;100,0;122,1;123,0;225,1;229,0;252,1;256,0;347,1;351,0;371,1;372,0;401,-1; 2:10,13,9,Times,0,12,0,0,0;9,13,9,Times,1,12,0,0,0; :[font = input; preserveAspect; startGroup] 1.) 5 x1 + 5 x2 + 2 x3 + 3 x4 + 7 x5 - x6 = 1 6 x1 + 2 x2 + 7 x3 + 7 x4 + 3 x5 + x6 = 17 9 x 1 + 8 x2 + 10 x3 + 6 x4 + 4 x5 - 3x6 = 2 8 x1 + 4 x2 + 2 x3 + 5 x4 + x5 = 4 3 x 1 + 10 x2 + 2 x3 + 7 x4 + x5 + 7x6 = 10 ;[s] 1:0,0;445,-1; 1:1,16,12,Times,0,14,0,0,0; :[font = message; inactive; preserveAspect; endGroup] Syntax::bktx: "1." has extra ")" after it. :[font = input; preserveAspect; startGroup] MatrixForm[M = {{5, 5, 2, 3, 7, -1}, {6,2,7,7,3, 1}, {9, 8, 10, 6, 4, -3}, {8, 4, 2, 5, 1, 0}, {3, 10, 2, 7, 1, 7}} ] ;[s] 4:0,0;15,2;161,0;164,1;177,-1; 3:2,13,10,Courier,1,12,0,0,0;1,18,13,Courier,1,18,0,0,0;1,13,9,Times,0,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{5, 5, 2, 3, 7, -1}, {6, 2, 7, 7, 3, 1}, {9, 8, 10, 6, 4, -3}, {8, 4, 2, 5, 1, 0}, {3, 10, 2, 7, 1, 7}}] ;[o] 5 5 2 3 7 -1 6 2 7 7 3 1 9 8 10 6 4 -3 8 4 2 5 1 0 3 10 2 7 1 7 :[font = input; preserveAspect] b = {1,17,2,4,10}; x = {x1, x2, x3, x4, x5, x6}; :[font = input; preserveAspect; startGroup] Print[MatrixForm[M.x]," ", MatrixForm[b]] :[font = print; inactive; preserveAspect; endGroup] 5 x1 + 5 x2 + 2 x3 + 3 x4 + 7 x5 - x6 1 6 x1 + 2 x2 + 7 x3 + 7 x4 + 3 x5 + x6 17 9 x1 + 8 x2 + 10 x3 + 6 x4 + 4 x5 - 3 x6 2 8 x1 + 4 x2 + 2 x3 + 5 x4 + x5 4 3 x1 + 10 x2 + 2 x3 + 7 x4 + x5 + 7 x6 10 :[font = input; preserveAspect; startGroup] MatrixForm[ AugM = Transpose[ Join[Transpose[M],{b}] ] ] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{5, 5, 2, 3, 7, -1, 1}, {6, 2, 7, 7, 3, 1, 17}, {9, 8, 10, 6, 4, -3, 2}, {8, 4, 2, 5, 1, 0, 4}, {3, 10, 2, 7, 1, 7, 10}}] ;[o] 5 5 2 3 7 -1 1 6 2 7 7 3 1 17 9 8 10 6 4 -3 2 8 4 2 5 1 0 4 3 10 2 7 1 7 10 :[font = input; preserveAspect; startGroup] MatrixForm[ RowReduce[ AugM ]]MatrixForm[ RowReduce[ AugM ]] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 0, 0, 0, 0, -941/1310, -811/655}, {0, 1, 0, 0, 0, 203/1310, -842/655}, {0, 0, 1, 0, 0, -61/131, -7/131}, {0, 0, 0, 1, 0, 3247/2620, 4927/1310}, {0, 0, 0, 0, 1, -363/2620, 457/1310}}] ;[o] 941 811 -(----) -(---) 1 0 0 0 0 1310 655 203 842 ---- -(---) 0 1 0 0 0 1310 655 61 7 -(---) -(---) 0 0 1 0 0 131 131 3247 4927 ---- ---- 0 0 0 1 0 2620 1310 363 457 -(----) ---- 0 0 0 0 1 2620 1310 :[font = input; preserveAspect] :[font = input; preserveAspect] Solution: :[font = input; preserveAspect] x1 = -811/655 + -941/1310 x6 x2 = -842/655 - 203/1310 x6 x3 = -7/131 + 61/131 x6 x4 = 4927/1310 - 3247/2620 x6 x5 = 457/1310 + 363/2620 x6 :[font = input; preserveAspect] 2.) 5 x1 + 5x2 + 2 x3 + 3 x4 = 0 6 x1 + 2x2 + 7 x3 + 7 x4 = 0 9 x 1 + 8 x2 + 10 x3 + 6 x4 = 0 ;[s] 2:0,0;11,1;171,-1; 2:1,13,10,Courier,0,12,0,0,0;1,16,12,Times,0,14,0,0,0; :[font = input; preserveAspect; startGroup] MatrixForm[M = {{5, 5, 2, 3}, {6,2,7,7}, {9, 8, 10, 6}} ] ;[s] 3:0,0;15,1;79,0;83,-1; 2:2,13,10,Courier,1,12,0,0,0;1,13,9,Times,0,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{5, 5, 2, 3}, {6, 2, 7, 7}, {9, 8, 10, 6}}] ;[o] 5 5 2 3 6 2 7 7 9 8 10 6 :[font = input; preserveAspect] b = {0, 0, 0}; x = {x1, x2, x3, x4}; ;[s] 2:0,0;36,1;37,-1; 2:1,13,10,Courier,1,12,0,0,0;1,18,13,Courier,1,18,0,0,0; :[font = input; preserveAspect; startGroup] Print[MatrixForm[M.x]," ", MatrixForm[b]] :[font = print; inactive; preserveAspect; endGroup] 5 x1 + 5 x2 + 2 x3 + 3 x4 0 6 x1 + 2 x2 + 7 x3 + 7 x4 0 9 x1 + 8 x2 + 10 x3 + 6 x4 0 :[font = input; preserveAspect; startGroup] MatrixForm[ AugM = Transpose[ Join[Transpose[M],{b}] ] ] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{5, 5, 2, 3, 0}, {6, 2, 7, 7, 0}, {9, 8, 10, 6, 0}}] ;[o] 5 5 2 3 0 6 2 7 7 0 9 8 10 6 0 :[font = input; preserveAspect; startGroup] MatrixForm[ RowReduce[ AugM ]] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 0, 0, 32/21, 0}, {0, 1, 0, -19/21, 0}, {0, 0, 1, -1/21, 0}}] ;[o] 32 -- 1 0 0 21 0 19 -(--) 0 1 0 21 0 1 -(--) 0 0 1 21 0 :[font = input; preserveAspect] Solution: x1 = -32/21 x4 x2 = 19/21 x4 x3 = 1/21 x4 :[font = input; preserveAspect] 3.) x1 + 2x2 - 3x3 + 4x4 = -5 6x1 - 7x2 + 8x3 + 9x4 = 10 11x1 - 12x2 12x4 = 3 3x1 - x2 + 17x3 = -9 ;[s] 1:0,0;176,-1; 1:1,13,10,Courier,0,12,0,0,0; :[font = input; preserveAspect; startGroup] MatrixForm[M = {{1, 2, -3, 4}, {6, -7, 8, 9}, {11, -12, 0, 12}, {3, -1, 17, 0}} ] ;[s] 3:0,0;15,1;114,0;118,-1; 2:2,13,10,Courier,1,12,0,0,0;1,13,9,Times,0,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 2, -3, 4}, {6, -7, 8, 9}, {11, -12, 0, 12}, {3, -1, 17, 0}}] ;[o] 1 2 -3 4 6 -7 8 9 11 -12 0 12 3 -1 17 0 :[font = input; preserveAspect] b = {-5, 10, 3, -9}; x = {x1, x2, x3, x4}; :[font = input; preserveAspect; startGroup] Print[MatrixForm[M.x]," ", MatrixForm[b]] :[font = print; inactive; preserveAspect; endGroup] x1 + 2 x2 - 3 x3 + 4 x4 -5 6 x1 - 7 x2 + 8 x3 + 9 x4 10 11 x1 - 12 x2 + 12 x4 3 3 x1 - x2 + 17 x3 -9 :[font = input; preserveAspect; startGroup] MatrixForm[ AugM = Transpose[ Join[Transpose[M],{b}] ] ] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 2, -3, 4, -5}, {6, -7, 8, 9, 10}, {11, -12, 0, 12, 3}, {3, -1, 17, 0, -9}}] ;[o] 1 2 -3 4 -5 6 -7 8 9 10 11 -12 0 12 3 3 -1 17 0 -9 :[font = input; preserveAspect; startGroup] MatrixForm[ RowReduce[ AugM ]] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 0, 0, 0, -6021/1091}, {0, 1, 0, 0, -10894/3273}, {0, 0, 1, 0, 814/3273}, {0, 0, 0, 1, 6482/3273}}] ;[o] 6021 -(----) 1 0 0 0 1091 10894 -(-----) 0 1 0 0 3273 814 ---- 0 0 1 0 3273 6482 ---- 0 0 0 1 3273 :[font = input; preserveAspect; startGroup] LinearSolve[M,b] :[font = output; output; inactive; preserveAspect; endGroup] {-6021/1091, -10894/3273, 814/3273, 6482/3273} ;[o] 6021 10894 814 6482 {-(----), -(-----), ----, ----} 1091 3273 3273 3273 :[font = input; preserveAspect] Solution: x1 = -6021/1091 x2 = -10894/3273 x3 = 814/3273 x4 = 6482/3273 :[font = input; preserveAspect; startGroup] MatrixForm[M.M] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-8, 20, 81, -14}, {79, -44, 79, 57}, {-25, 94, 75, -64}, {184, -191, -17, 207}}] ;[o] -8 20 81 -14 79 -44 79 57 -25 94 75 -64 184 -191 -17 207 :[font = input; preserveAspect; startGroup] MatrixForm[Inverse[M]] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{168/1091, -396/1091, 241/1091, 216/1091}, {249/1091, -436/3273, 26/1091, 337/3273}, {-15/1091, 184/3273, -41/1091, 98/3273}, {95/1091, 653/3273, -104/1091, -257/3273}}] ;[o] 168 396 241 216 ---- -(----) ---- ---- 1091 1091 1091 1091 249 436 26 337 ---- -(----) ---- ---- 1091 3273 1091 3273 15 184 41 98 -(----) ---- -(----) ---- 1091 3273 1091 3273 95 653 104 257 ---- ---- -(----) -(----) 1091 3273 1091 3273 :[font = input; preserveAspect] 4.) x1 + 2x2 - 3x3 + 4x4 = 0 6x1 - 7x2 + 8x3 + 9x4 = 0 11x1 - 12x2 12x4 = 0 3x1 - x2 + 17x3 = 0 ;[s] 1:0,0;172,-1; 1:1,13,10,Courier,0,12,0,0,0; :[font = input; preserveAspect; startGroup] MatrixForm[M = {{1, 2, -3, 4}, {6, -7, 8, 9}, {11, -12, 0, 12}, {3, -1, 17, 0}} ] ;[s] 3:0,0;15,1;114,0;118,-1; 2:2,13,10,Courier,1,12,0,0,0;1,13,9,Times,0,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 2, -3, 4}, {6, -7, 8, 9}, {11, -12, 0, 12}, {3, -1, 17, 0}}] ;[o] 1 2 -3 4 6 -7 8 9 11 -12 0 12 3 -1 17 0 :[font = input; preserveAspect] b = {0, 0, 0, 0}; x = {x1, x2, x3, x4}; :[font = input; preserveAspect; startGroup] Print[MatrixForm[M.x]," ", MatrixForm[b]] :[font = print; inactive; preserveAspect; endGroup] x1 + 2 x2 - 3 x3 + 4 x4 0 6 x1 - 7 x2 + 8 x3 + 9 x4 0 11 x1 - 12 x2 + 12 x4 0 3 x1 - x2 + 17 x3 0 :[font = input; preserveAspect; startGroup] MatrixForm[ AugM = Transpose[ Join[Transpose[M],{b}] ] ] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 2, -3, 4, 0}, {6, -7, 8, 9, 0}, {11, -12, 0, 12, 0}, {3, -1, 17, 0, 0}}] ;[o] 1 2 -3 4 0 6 -7 8 9 0 11 -12 0 12 0 3 -1 17 0 0 :[font = input; preserveAspect; startGroup] MatrixForm[ RowReduce[ AugM ]] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}] ;[o] 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 :[font = input; preserveAspect; startGroup] LinearSolve[M,b] :[font = output; output; inactive; preserveAspect; endGroup] {0, 0, 0, 0} ;[o] {0, 0, 0, 0} :[font = input; preserveAspect] Solution: x1 = 0 x2 = 0 x3 = 0 x4 = 0 :[font = input; preserveAspect; startGroup] MatrixForm[M.M] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-8, 20, 81, -14}, {79, -44, 79, 57}, {-25, 94, 75, -64}, {184, -191, -17, 207}}] ;[o] -8 20 81 -14 79 -44 79 57 -25 94 75 -64 184 -191 -17 207 :[font = input; preserveAspect; startGroup] MatrixForm[Inverse[M]] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{168/1091, -396/1091, 241/1091, 216/1091}, {249/1091, -436/3273, 26/1091, 337/3273}, {-15/1091, 184/3273, -41/1091, 98/3273}, {95/1091, 653/3273, -104/1091, -257/3273}}] ;[o] 168 396 241 216 ---- -(----) ---- ---- 1091 1091 1091 1091 249 436 26 337 ---- -(----) ---- ---- 1091 3273 1091 3273 15 184 41 98 -(----) ---- -(----) ---- 1091 3273 1091 3273 95 653 104 257 ---- ---- -(----) -(----) 1091 3273 1091 3273 :[font = input; preserveAspect] 5.) x1 + 2x2 - 3x3 + 4x4 = -5 6x1 - 7x2 + 8x3 + 9x4 = 10 11x1 - 12x2 12x4 = 3 4x1 - 7x2 - 5x3 - x4 = -3 ;[s] 4:0,0;1,1;8,0;10,2;177,-1; 3:2,13,10,Courier,1,12,0,0,0;1,13,9,Times,0,12,0,0,0;1,13,10,Courier,0,12,0,0,0; :[font = input; preserveAspect; startGroup] MatrixForm[M = {{1, 2, -3, 4}, {6, -7, 8, 9}, {11, -12, 0, 12}, {4, -7, -5, -1}} ] ;[s] 3:0,0;15,1;115,0;118,-1; 2:2,13,10,Courier,1,12,0,0,0;1,13,9,Times,0,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 2, -3, 4}, {6, -7, 8, 9}, {11, -12, 0, 12}, {4, -7, -5, -1}}] ;[o] 1 2 -3 4 6 -7 8 9 11 -12 0 12 4 -7 -5 -1 :[font = input; preserveAspect] b = {4, 9, 12, -1}; x = {x1, x2, x3, x4}; :[font = input; preserveAspect; startGroup] Print[MatrixForm[M.x]," ", MatrixForm[b]] :[font = print; inactive; preserveAspect; endGroup] x1 + 2 x2 - 3 x3 + 4 x4 4 6 x1 - 7 x2 + 8 x3 + 9 x4 9 11 x1 - 12 x2 + 12 x4 12 4 x1 - 7 x2 - 5 x3 - x4 -1 :[font = input; preserveAspect; startGroup] MatrixForm[ AugM = Transpose[ Join[Transpose[M],{b}] ] ] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 2, -3, 4, 4}, {6, -7, 8, 9, 9}, {11, -12, 0, 12, 12}, {4, -7, -5, -1, -1}}] ;[o] 1 2 -3 4 4 6 -7 8 9 9 11 -12 0 12 12 4 -7 -5 -1 -1 :[font = input; preserveAspect; startGroup] MatrixForm[ RowReduce[ AugM ]] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 0, 0, 648/257, 648/257}, {0, 1, 0, 337/257, 337/257}, {0, 0, 1, 98/257, 98/257}, {0, 0, 0, 0, 0}}] ;[o] 648 648 --- --- 1 0 0 257 257 337 337 --- --- 0 1 0 257 257 98 98 --- --- 0 0 1 257 257 0 0 0 0 0 :[font = input; preserveAspect] Solution: x1 = 648/257 - 648/257 x4 x2 = 337/257 - 337/257 x4 x3 = 98/257 - 98/257 x4 :[font = input; preserveAspect; startGroup] MatrixForm[M.M] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] MatrixForm[{{-4, -4, -7, -18}, {88, -98, -119, 48}, {-13, 22, -189, -76}, {-97, 124, -63, -106}}] ;[o] -4 -4 -7 -18 88 -98 -119 48 -13 22 -189 -76 -97 124 -63 -106 ^*)