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Notebook[{

Cell[CellGroupData[{
Cell["Row Reduction and Arithmetic", "Title"],

Cell["February 15, 2006", "Subtitle"],

Cell[CellGroupData[{

Cell["Some Special Matrices", "Subsubtitle"],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " has built-in commands for some common matrices. Play with (that means \
change the arguments and see how the ouput changes) the following commands \
until you understand how they work. They will save you some typing."
}], "Text"],

Cell[BoxData[
    \(MatrixForm[IdentityMatrix[5]]\)], "Input"],

Cell[BoxData[
    \(MatrixForm[DiagonalMatrix[{1.0, 2.2, \(-1.1\)}]]\)], "Input"],

Cell[BoxData[
    \(MatrixForm[Table[Random[], {i, 5}, {j, 7}]]\)], "Input"],

Cell[TextData[{
  "The command \"Random[]\" produces a random number between 0 and 1. Use ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s help system to find the command \"Random\", and look at the examples \
given there. Now produce a matrix with 3 rows and 5 columns whose entries are \
random real numbers between 1.1 and 22."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Row Reduction", "Subsubtitle"],

Cell[TextData[{
  "The command \"Timing[ whatever; ]\" tells you how long it took ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to evaluate \"whatever\". For example:"
}], "Text"],

Cell[BoxData[
    \(Timing[\(\[Sum]\+\(i = 1\)\%1000000 i\^3;\)]\)], "Input"],

Cell[TextData[{
  "On my laptop the output of this is \"{11.897 Second,Null}\". This tells us \
that it takes ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " about 12 seconds to compute the sum of the cubes of the first million \
natural numbers, and the \"Null\" happened because of the semicolon (remove \
the semicolon and rerun if that didn't make sense)."
}], "Text"],

Cell[TextData[{
  "How long does it take ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to find the reduced row echelon form of a 5\[Cross]7 matrix, with random \
real entries between 0 and 1? The command to get the reduced row echelon form \
is \"RowReduce[ matrix ]\"."
}], "Text"],

Cell["How long did it take for you to type in that command?", "Text"],

Cell[TextData[{
  "How big does the matrix have to be before it takes ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " 10 seconds to find the reduced row echelon form? How big does it have to \
be before it takes 100 seconds?"
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Arithmetic", "Subsubtitle"],

Cell["Here's how to add matrices, and some other things.", "Text"],

Cell[BoxData[{
    \(MatrixForm[
      A = Table[
          Random[Integer, {1, 5}], {i, 3}, {j, 
            5}]]\), "\[IndentingNewLine]", 
    \(MatrixForm[
      B = Table[
          Random[Integer, {1, 5}], {i, 3}, {j, 
            5}]]\), "\[IndentingNewLine]", 
    \(MatrixForm[A + B]\), "\[IndentingNewLine]", 
    \(MatrixForm[\(1\/3\) A]\), "\[IndentingNewLine]", 
    \(MatrixForm[Transpose[A]]\)}], "Input"],

Cell["Multiplying matrices is trickier.", "Text"],

Cell[BoxData[{
    \(MatrixForm[
      M1 = Table[
          Random[Integer, {\(-10\), 10}], {i, 3}, {j, 
            3}]]\), "\[IndentingNewLine]", 
    \(MatrixForm[
      M2 = Table[
          Random[Integer, {\(-10\), 10}], {i, 3}, {j, 
            3}]]\), "\[IndentingNewLine]", 
    \(MatrixForm[M1 . M2]\)}], "Input"],

Cell[TextData[{
  "That's a period between the two matrices. ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " will not let you use \"C\", \"D\", or \"N\" as variables because they are \
commands (\"N\" converts an expression to a decimal, \"D\" takes a \
derivative, \"C\" is an integration constant). What happens if you try to \
multiply two matrices with incompatible shapes?"
}], "Text"],

Cell["We also learned about the inverse of a square matrix.", "Text"],

Cell[BoxData[{
    \(MatrixForm[
      A = Table[
          Random[Integer, {\(-5\), 5}], {i, 3}, {j, 
            3}]]\), "\[IndentingNewLine]", 
    \(MatrixForm[B = Inverse[A]]\), "\[IndentingNewLine]", 
    \(MatrixForm[A . B]\), "\[IndentingNewLine]", 
    \(MatrixForm[B . A]\), "\[IndentingNewLine]", 
    \(\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Challenge Problem", "Subsubtitle"],

Cell["\<\
Let's start by taking a 3\[Cross]5 matrix, and then think about how Gaussian \
Elimination works.\
\>", "Text"],

Cell[BoxData[
    RowBox[{
      RowBox[{"A", "=", 
        TagBox[
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                {"0", "3", "5", \(-1\), "4"},
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              RowSpacings->1,
              ColumnSpacings->1,
              ColumnAlignments->{Left}], "\[NoBreak]", ")"}],
          Function[ BoxForm`e$, 
            MatrixForm[ BoxForm`e$]]]}], ";"}]], "Input"],

Cell["\<\
The first step is to interchange the top two rows. Find a 3\[Cross]3 matrix \
R1 so that R1.A is just like A, but with the first two rows exchanged.\
\>", "Text"],

Cell[BoxData[{
    RowBox[{
      RowBox[{"R1", "=", 
        RowBox[{"(", GridBox[{
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              {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]"}
              }], ")"}]}], ";"}], "\[IndentingNewLine]", \(MatrixForm[
      R1 . A]\)}], "Input"],

Cell["\<\
Now you need to multiply the first row by -1. Find a 3\[Cross]3 matrix R2 so \
that R2.(R1.A) is just like R1.A, but with the first row multiplied by -1.\
\>", "Text"],

Cell[BoxData[{
    RowBox[{
      RowBox[{"R2", "=", 
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      R2 . \((R1 . A)\)]\)}], "Input"],

Cell["\<\
The next step is to replace the third row with (third row-2*first row). Find \
a 3\[Times]3 matrix R3 so that R3.(R2.(R1.A)) is just like (R2.(R1.A)), but \
with the third row replaced by  (third row-2*first row).\
\>", "Text"],

Cell[BoxData[{
    RowBox[{
      RowBox[{"R3", "=", 
        RowBox[{"(", GridBox[{
              {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]"},
              {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]"},
              {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]"}
              }], ")"}]}], ";"}], "\[IndentingNewLine]", \(MatrixForm[
      R3 . \((R2 . \((R1 . A)\))\)]\)}], "Input"],

Cell["Do the parentheses matter?", "Text"],

Cell["Can you complete the Gaussian elimination like this?", "Text"]
}, Open  ]]
}, Open  ]]
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(*******************************************************************
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