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Cell[CellGroupData[{
Cell["Sample Quiz", "Title"],

Cell["\<\
March 8, 2006
MTH 338: Linear Algebra\
\>", "Subtitle"],

Cell[TextData[{
  "The determinant of ",
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  " is ______________________"
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  " is _____________________"
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  "The reduced row echelon form of the matrix ",
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  " is ",
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  "."
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Cell[TextData[{
  "The definition of the determinant is as follows:\n\tDet([a])  = a,\n\tand \
if A is an n\[Cross]n matrix, then for any ",
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  " between 1 and ",
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  " (inclusive)\n\tDet(A) = ",
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  ", where ",
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  " and ",
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  " is the matrix obtained from ",
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  " by deleting the ",
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  "-th row and the ",
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  "-th column.\n\t\nUsing ",
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  "We give the proof using mathematical induction, inducting on the size of \
the matrix. If ",
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  " is a 1\[Cross]1 matrix, then \"if two rows are identical, then det=0\" \
has a false hypothesis, so is true. If two rows of a 2\[Cross]2 matrix are \
identical, then the matrix looks like ",
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  ", so the statement is true for 2\[Cross]2 matrices.\n\nNow suppose that \
\"if two rows of an n\[Cross]n matrix are identical, then its determinant is \
zero\". We call this the induction hypothesis, and we need to show that \"if \
two rows of an (n+1)\[Cross](n+1) matrix are identical, then its determinant \
is zero\". Let ",
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  " be an  (n+1)\[Cross](n+1) matrix with two identical rows, and let ",
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  " be the number of a row which is not one the two identical ones. We have \
by the definition of determinant:\n\t",
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  "-th row and ",
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  " for every ",
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  ". Thus\n\t",
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          A] = \(\(\(a\_\(r\ 1\)\) A\_\(r\ 1\) + \(a\_\(r\ 2\)\) 
                  A\_\(r\ 2\) + \  ... \)\  + \(a\_\(r\ n\)\) 
              A\_\(r\ n\) = \(\(\(a\_\(r\ 1\)\) 0 + \(a\_\(r\ 2\)\) 
                    0 + \  ... \)\  + \(a\_\(r\ n\)\) 0\  = 0\)\)\)]],
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