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Linear Algebra For Dummies
Part III: Evaluating Determinants .............................. 173
Chapter 9: Keeping Things in Order with Permutations . . . . . . . . . . .175
Computing and Investigating Permutations .............................................176
Counting on finding out how to count ............................................176
Making a list and checking it twice ..................................................177
Bringing permutations into matrices
(or matrices into permutations) ..................................................180
Involving Inversions in the Counting ........................................................181
Investigating inversions ....................................................................181
Inviting even and odd inversions to the party ...............................183
Chapter 10: Determining Values of Determinants . . . . . . . . . . . . . . . .185
Evaluating the Determinants of 2 × 2 Matrices ........................................185
Involving permutations in determining the determinant .............186
Coping with cofactor expansion ......................................................189
Using Determinants with Area and Volume .............................................192
Finding the areas of triangles ...........................................................192
Pursuing parallelogram areas ..........................................................195
Paying the piper with volumes of parallelepipeds ........................198
Chapter 11: Personalizing the Properties of Determinants . . . . . . . .201
Transposing and Inverting Determinants .................................................202
Determining the determinant of a transpose .................................202
Investigating the determinant of the inverse .................................203
Interchanging Rows or Columns ................................................................204
Zeroing In on Zero Determinants ..............................................................206
Finding a row or column of zeros ....................................................206
Zeroing out equal rows or columns ................................................206
Manipulating Matrices by Multiplying and Combining ...........................209
Multiplying a row or column by a scalar ........................................209
Adding the multiple of a row or column
to another row or column .............................................................212
Taking on Upper or Lower Triangular Matrices ......................................213
Tracking down determinants of triangular matrices ....................213
Cooking up a triangular matrix from scratch .................................214
Creating an upper triangular or lower triangular matrix ..............217
Determinants of Matrix Products ..............................................................221
Chapter 12: Taking Advantage of Cramer’s Rule . . . . . . . . . . . . . . . . .223
Inviting Inverses to the Party with Determined Determinants ..............223
Setting the scene for finding inverses .............................................224
Introducing the adjoint of a matrix..................................................225
Instigating steps for the inverse ......................................................228
Taking calculated steps with variable elements ............................229