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Reduced-Row Echelon Form
The set of equations
x – 2y + 3z = 14
2x + y – z = -3
4x – 2y + 2z = 14
can be written as the augmented matrix
Which can then be stored as a 3 x 4 real matrix in any matrix variable. M1 is used in this example.
You can then use the RREF function to change this to reduced-row echelon form, storing it in any matrix
variable. M2 is used in this example.
The reduced row echelon matrix gives the solution to the linear equation in the fourth column.
An advantage of using the RREF function is that it will also work with inconsistent matrices resulting from
systems of equations which have no solution or innite solutions.
542 Chapter 26 Matrices
- HP Prime Graphing Calculator 1
- Table of contents 3
- 1 Preface 49
- 2 Getting started 50
- On/o, cancel operations 51
- The display 52
- The Quick Settings menu 53
- Navigation 54
- Touch gestures 55
- The keyboard 55
- Context-sensitive menu 56
- Entry and edit keys 57
- Shift keys 58
- Adding text 59
- Math keys 59
- Math template 61
- Math shortcuts 62
- Fractions 63
- Hexagesimal numbers 63
- EEX key (powers of 10) 64
- Selecting from a menu 65
- Input forms 66
- System-wide settings 67
- Home settings 68
- System-wide settings 21 69
- Specifying a Home setting 70
- Mathematical calculations 71
- Where to start 72
- Choosing an entry type 72
- Entering expressions 72
- Parentheses 73
- Negative numbers 74
- Large results 75
- Using the clipboard 75
- Reusing the last result 76
- Storing value in a variable 77
- Complex numbers 78
- Copy and paste 79
- Sharing data 81
- Using Memory Manager 82
- Online help 84
- History in RPN mode 87
- Reusing results 88
- Sample calculations 89
- Manipulating the stack 90
- Show an item 93
- Delete an item 93
- Delete all items 94
- CAS view 95
- CAS calculations 96
- Example 1 97
- Settings 98
- Settings 51 99
- Using a Home variable in CAS 101
- 5 Exam Mode 102
- Creating a new conguration 105
- Activating exam mode 106
- Modifying congurations 108
- Deleting congurations 109
- 6 Introduction to HP apps 110
- Application library 111
- App views 112
- Symbolic view 113
- Symbolic Setup view 114
- Plot view 114
- Plot Setup view 115
- Numeric view 115
- Numeric Setup view 116
- Quick example 117
- Adding a denition 120
- Modifying a denition 121
- Denitional building blocks 121
- Choosing a color for plots 123
- Deleting a denition 123
- Press 124
- Restoring default settings 125
- Zoom factors 126
- Zoom options 127
- Zoom gestures 127
- Zoom keys 127
- Zoom menu 127
- Box zoom 128
- Views menu 129
- Zoom examples 130
- Autoscale 135
- Autoscale 135
- Selecting a plot 137
- Evaluating a function 138
- Turning tracing on or o 138
- Conguring Plot view 139
- Graphing methods 142
- Evaluating 147
- Custom tables 148
- Deleting data 149
- Copying and pasting a cell 149
- Copying and pasting a row 149
- More menu 151
- Adding a note to an app 153
- Creating an app 153
- App functions and variables 155
- Qualifying variables 156
- 7 Function app 157
- Dening the expressions 158
- Setting up a plot 159
- Plotting a function 159
- Tracing a graph 160
- Changing the scale 161
- Displaying Numeric view 161
- Setting up Numeric view 163
- Exploring Numeric view 164
- Analyzing functions 165
- Displaying the Plot view menu 166
- Modifying function graphs 167
- Function variables 175
- – x – Root 176
- 128 Chapter 7 Function app 176
- Summary of FCN operations 177
- Functions dened by integrals 180
- 8 Advanced Graphing app 182
- Dening an open sentence 185
- Setting up the plot 185
- Exploring the graph 187
- Trace in Plot view 190
- Trace in Numeric view 193
- Plot Gallery 196
- 9 Geometry 197
- Adding a constrained point 198
- Adding a tangent 199
- Creating a derivative point 200
- Adding some calculations 202
- Calculations in Plot view 204
- Trace the derivative 204
- Plot view in detail 205
- Selecting objects 206
- Hiding names 206
- Moving objects 206
- Coloring objects 207
- Filling objects 207
- Clearing an object 207
- Clearing all objects 208
- Gestures in Plot view 208
- Plot view: buttons and keys 209
- The Options menu 210
- Using the slopeeld command 210
- Symbolic view in detail 212
- Creating objects 213
- Reordering entries 213
- Hiding an object 213
- Numeric view in detail 214
- Listing all objects 217
- Editing a calculation 217
- Plot view: Cmds menu 218
- Angle bisector 221
- Triangle 221
- Isosceles Triangle 221
- Right Triangle 222
- Quadrilateral 222
- Parallelogram 222
- Rectangle 222
- Regular Polygon 223
- Circumcircle 224
- Excircle 224
- Incircle 224
- Hyperbola 225
- Parabola 225
- Function 227
- Parametric 227
- Sequence 228
- Implicit 228
- Slopeeld 228
- Transform 229
- Reection 230
- Rotation 231
- Dilation 231
- Similarity 232
- Projection 232
- Inversion 233
- Reciprocation 233
- Cartesian 234
- Symbolic view: Cmds menu 237
- Numeric view: Cmds menu 248
- Other Geometry functions 252
- 10 Spreadsheet 259
- Basic operations 264
- Cell naming 265
- Entering content 265
- Direct entry 266
- Importing data 267
- External functions 267
- Using the CHOOSE command 269
- External references 269
- Referencing variables 270
- Buttons and keys 271
- Formatting options 272
- Format parameters 273
- Spreadsheet functions 274
- 11 Statistics 1Var app 275
- Symbolic view: menu items 278
- Numeric view: menu items 282
- Editing a data set 283
- Inserting data 283
- Generating data 283
- Computed statistics 284
- Plotting 285
- Plot types 286
- Box-and-whisker plot 287
- Normal probability plot 287
- Line plot 288
- Bar graph 288
- Pareto chart 289
- Control chart 289
- Dot plot 290
- Stem and leaf plot 290
- Plot view: menu items 292
- 12 Statistics 2Var app 293
- Entering data 294
- Choosing data columns and t 295
- Exploring statistics 296
- Plotting the graph 298
- Displaying the equation 299
- Predicting values 299
- Dening a regression model 301
- Choosing a t 302
- Fit types 302
- Dening your own t 302
- Plotting statistical data 304
- Tracing a scatter plot 305
- Tracing a curve 305
- Tracing order 306
- Troubleshooting a plot 308
- 13 Inference app 309
- Opening the Inference app 310
- Symbolic view options 310
- Table 13-3 X 311
- Table 13-4 Regression 311
- Table 13-5 ANOVA 311
- Displaying the test results 314
- Plotting the test results 314
- Importing statistics 315
- Calculating the statistics 317
- Importing the data 318
- Importing statistics 271 319
- Hypothesis tests 320
- One-Sample Z-Test 321
- Two-Sample Z-Test 321
- One-Proportion Z-Test 322
- Two-Proportion Z-Test 323
- One-Sample T-Test 325
- Two-Sample T-Test 326
- Condence intervals 327
- Two-Sample Z-Interval 328
- One-Proportion Z-Interval 329
- Two-Proportion Z-Interval 329
- One-Sample T-Interval 330
- Two-Sample T-Interval 331
- Chi-square tests 332
- Two-way table test 333
- Inference for regression 334
- Condence interval for slope 335
- Menu name 336
- Menu keys 336
- Prediction interval 338
- 14 Solve app 341
- Opening the Solve app 342
- Entering known variables 343
- Plotting the equation 344
- Several equations 345
- Opening the solve app 346
- Dening the equations 346
- Entering a seed value 346
- Limitations 347
- Solution information 348
- 15 Linear Solver app 349
- Menu items 351
- 16 Parametric app 352
- Dening the functions 353
- Setting the angle measure 353
- Plotting the functions 355
- Displaying the Numeric view 356
- 17 Polar app 357
- Plotting the expression 359
- 18 Sequence app 362
- Opening the Sequence app 363
- Dening the expression 363
- Plotting the sequence 365
- Exploring the table of values 367
- 19 Finance app 371
- Cash ow diagrams 373
- 326 Chapter 19 Finance app 374
- Time value of money (TVM) 375
- Amortizations 377
- 330 Chapter 19 Finance app 378
- Amortization graph 379
- 20 Triangle Solver app 380
- Specify the known values 381
- Choosing triangle types 382
- Special cases 383
- Not enough data 384
- 21 The Explorer apps 385
- Graph mode 386
- Equation mode 386
- Test mode 387
- Quadratic Explorer app 388
- Trig Explorer app 390
- 22 Functions and commands 393
- Keyboard functions 395
- Math menu 400
- Arithmetic 401
- Arithmetic – Complex 402
- Arithmetic – Exponential 403
- Trigonometry 403
- Hyperbolic 404
- TANH(value) 405
- ATANH(value) 405
- Math menu 357 405
- Probability 406
- Probability – Random 406
- Probability – Density 407
- Probability – Cumulative 408
- Binomial 409
- Geometric 409
- Probability – Inverse 410
- CAS menu 412
- Algebra – Extract 414
- Calculus 414
- Integrate 415
- Summation 415
- Calculus – Dierential 416
- Calculus – Integral 416
- Calculus – Limits 417
- Calculus – Transform 418
- Complex Solve 419
- Complex Zeros 419
- Numerical Solve 420
- Dierential Equation 420
- ODE Solve 420
- Linear System 420
- Rewrite – Exp & Ln 421
- Rewrite – Sine 422
- Rewrite – Cosine 422
- Rewrite – Tangent 423
- Rewrite – Trig 424
- Divisors 425
- Factor List 425
- Integer – Prime 426
- Integer – Division 427
- Polynomial 428
- Polynomial – Create 429
- Polynomial – Algebra 430
- Polynomial – Special 431
- Groebner Remainder 432
- Lagrange 432
- Laguerre 432
- Legendre 432
- App menu 433
- Function app functions 434
- Solve app functions 435
- Spreadsheet app functions 435
- HypZ1mean 442
- HYPZ2mean 443
- HypZ1prop 443
- HypZ2prop 444
- HypT1mean 445
- HypT2mean 445
- ConfZ1mean 446
- ConfZ2mean 446
- ConfZ1prop 447
- ConfZ2prop 447
- ConfT1mean 447
- Statistics 1Var app functions 448
- Statistics 2Var app functions 449
- Inference app functions 450
- HypZ2mean 451
- ConfT2mean 455
- Chi2TwoWay 455
- LinRegrTConf- Slope 455
- LinRegrTConfInt 456
- LinRegrTMean-Resp 456
- LinRegrTPredInt 457
- LinRegrTTest 457
- Finance app functions 458
- Linear Solver app functions 459
- Triangle Solver app functions 460
- Linear Explorer functions 461
- Quadratic Explorer functions 462
- Common app functions 462
- Ctlg menu 464
- Day of the week 474
- LineVert 488
- PIECEWISE 495
- POLYROOT 497
- Creating your own functions 515
- 23 Variables 517
- Working with user variables 518
- Working with app variables 519
- More about the Vars menu 519
- Home variables 521
- App variables 522
- Results variables 523
- Geometry app variables 524
- Spreadsheet app variables 524
- Solve app variables 524
- App variables 477 525
- Statistics 1Var app variables 526
- Statistics 2Var app variables 528
- Inference app variables 531
- Parametric app variables 533
- Polar app variables 534
- Finance app variables 535
- Linear Solver app variables 535
- Triangle Solver app variables 536
- Linear Explorer app variables 536
- Trig Explorer app variables 536
- Sequence app variables 537
- 24 Units and constants 538
- Unit calculations 539
- Length. Select cm 540
- 5 inches rst 541
- Unit calculations 493 541
- Unit tools 543
- Physical constants 544
- List of constants 547
- 25 Lists 549
- Button or Key 550
- The List Editor 551
- Editing a list 552
- Deleting lists 554
- Lists in Home view 555
- To display a list 556
- To display one element 556
- To store one element 556
- List references 556
- To send a list 556
- List functions 557
- Intersect 558
- Make List 558
- Concatenate 559
- Position 559
- 5. Calculate the statistics 563
- Tap when you are done 563
- 26 Matrices 564
- Working with matrices 565
- Matrix Editor: More menu 566
- Matrices in Home view 568
- Storing a matrix 570
- Displaying a matrix 571
- Displaying one element 571
- Storing one element 571
- Matrix arithmetic 572
- 2, press the following keys: 574
- 526 Chapter 26 Matrices 574
- Multiplying two matrices 575
- Raising a matrix to a power 575
- Dividing by a square matrix 576
- Inverting a matrix 576
- Negating each element 577
- 530 Chapter 26 Matrices 578
- Matrix functions and commands 580
- Matrix functions 581
- Identity 582
- Isometric 583
- Vandermonde 583
- Row Norm 583
- Column Norm 584
- Spectral Norm 584
- Spectral Radius 584
- Condition 584
- Advanced 585
- Diagonal 586
- Cholesky 586
- Hessenberg 586
- Factorize 587
- Cross Product 588
- Dot Product 588
- Examples 589
- Reduced-Row Echelon Form 590
- 27 Notes and Info 592
- The Note Editor 593
- Creating a note for an app 594
- Note Editor: buttons and keys 594
- The Note Editor 547 595
- Text formatting 596
- The Note Editor 549 597
- To import a note 598
- 28 Programming in HP PPL 600
- The Program Catalog 601
- Creating a new program 602
- Creating a new program 555 603
- The Program Editor 604
- Creating a new program 557 605
- Button or Key Meaning 606
- Deletes the entire program 606
- Run a Program 609
- Debug a Program 611
- Edit a Program 613
- Delete a program 614
- Delete all programs 614
- Variables and visibility 615
- Program ROLLDIE 617
- Program ROLLMANY 617
- User mode 619
- Reassigning keys 620
- Key names 620
- App programs 624
- Redening the View menu 625
- Customizing an app 625
- Program commands 631
- Variable 636
- Commands under the Cmds menu 637
- String Result 639
- STRING(2/3) 0.666666666667 639
- Pixels and Cartesian 641
- App Functions 651
- Variables and programs 661
- 29 Basic integer arithmetic 687
- The default base 688
- Changing the default base 689
- Mixed-base arithmetic 689
- Integer manipulation 690
- Base functions 691
- 30 Appendix A – Glossary 693
- Calculator not responding 695
- Operating limits 695
- Status messages 695
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