2x2 Matrix

Regel von Sarrus (3x3)

Regel von Sarrus (3x3 Determinante)

Gauß'sches Eliminationsverfahren zur Determinantenberechnung

Determinantenberechnung mit Gauß'schem Eliminationsverfahren

Der Laplace'sche Entwicklungssatz

Unterschied In/Homogen

Basis, Basiswechsel

Abbildungsmatrizen

Funktionen bestimmen

Geraden und Ebenen

Gruppen

lineare abbildung 

quadratische ergänzung

Baustatik 1

Anfangswertprobleme :)

Differentialgleichungen

basis 

QR-Algorithmus

QR-Zerlegung

Cholesky-Zerlegung

Bild & Urbild

Zweigstromanalyse

Transistor als Thermometer

Investment

Teilfolgen

Epsilon Delta Kriterium

prinzip der virtuellen verrückung 

Linear Kombination

Folgen, Reihen und Potenzreihen

Flächenpressung

Arbeit 

Cholesky Zerlegung

Polynomfunktionen beschreiben

Zu Maß-und Integrationstheorie: Majorisierte Konvergenz

Teilchenphysik und Festkörperphysik

Orthogonaltrajektorien

test

Taylorreihen

Regel von de L'Hopital

Mathe

KRAFT

Grenzwert Funktion 

Folgenkriterium

ϵ-δ Kriterium 

Beweis Stetigkeit 

Wahrheitstabelle

Matrix invertieren

Basiswechselsatz

Lineare Abbildunge 

Vorderrad fahradgabel lagerkräfte berechnen 

Spannungsregelung

Jordan Normal Form

Test

Induktion 

Jordan-Normalform

Ansatz der Störfunktion zur Bestimmung der Partikulären Lösung

Kurvenintegral

dlg mithilfe inhomogenen linearen system lösen

Inverse einer Matrix

Gleichmäßige Stetigkeit

addition

Regel von L´Hospital

Lineare Abbildungen

dimension eines Untervektorraumes bestimmen

Stationäre Stoffübertragung

Polplan, Prinzip der virtuellen Verrückungen

komplexe Ungleichungen 

Bidualräume

Dualräume

Taylorpolynome

Kern und Basis

Diagonalisieren einer Matrix

Rekursive Folgen

Exponentialreihe

Darstellungsmatrizen

restglied

Basiswechsel

Gamma Funktion

Newton Verfahren

Randwertaufgabe

Numerische Integration: Trapez- und Simpsonregel

Funktionenfolgen

dualräume

Dualraum

Komplexe Nullstellen

Chemie

Natascha Kirchmann

Differentialquotient

Folgedefinition des Funktionslimes

Grenzwerte und Stetigkeit 

Komplexe Zahlen

Kern und Bild 

Direkte Summe 

Koordinaten 

Exponentielle Verzinsung

Lineare Verzinsung

Festverzinsliche Wertpapiere

<p>Es existiert für $A$ die Inverse $A^{-1}$ für alle Werte von $a \in \mathbb{R} \backslash\left\{\pm \frac{1}{\sqrt{2}}\right\}$</p>

<p>Es existiert für <mjx-container class="MathJax" jax="SVG"><svg alt="A" style="vertical-align: 0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-7249-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-7249-TEX-I-1D434"></use></g></g></g></svg></mjx-container> die Inverse <mjx-container class="MathJax" jax="SVG"><svg alt="A^{-1}" style="vertical-align: 0" xmlns="http://www.w3.org/2000/svg" width="3.854ex" height="1.887ex" role="img" focusable="false" viewBox="0 -833.9 1703.7 833.9" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-7250-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-7250-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-7250-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use xlink:href="#MJX-7250-TEX-I-1D434"></use></g><g data-mml-node="TeXAtom" transform="translate(750, 363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use xlink:href="#MJX-7250-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778, 0)"><use xlink:href="#MJX-7250-TEX-N-31"></use></g></g></g></g></g></svg></mjx-container> für alle Werte von <mjx-container class="MathJax" jax="SVG"><svg alt="a \in \mathbb{R} \backslash\left\{\pm \frac{1}{\sqrt{2}}\right\}" style="vertical-align: -1.469ex" xmlns="http://www.w3.org/2000/svg" width="14.666ex" height="4.07ex" role="img" focusable="false" viewBox="0 -1149.5 6482.3 1799" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-7251-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path id="MJX-7251-TEX-N-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path id="MJX-7251-TEX-D-211D" d="M17 665Q17 672 28 683H221Q415 681 439 677Q461 673 481 667T516 654T544 639T566 623T584 607T597 592T607 578T614 565T618 554L621 548Q626 530 626 497Q626 447 613 419Q578 348 473 326L455 321Q462 310 473 292T517 226T578 141T637 72T686 35Q705 30 705 16Q705 7 693 -1H510Q503 6 404 159L306 310H268V183Q270 67 271 59Q274 42 291 38Q295 37 319 35Q344 35 353 28Q362 17 353 3L346 -1H28Q16 5 16 16Q16 35 55 35Q96 38 101 52Q106 60 106 341T101 632Q95 645 55 648Q17 648 17 665ZM241 35Q238 42 237 45T235 78T233 163T233 337V621L237 635L244 648H133Q136 641 137 638T139 603T141 517T141 341Q141 131 140 89T134 37Q133 36 133 35H241ZM457 496Q457 540 449 570T425 615T400 634T377 643Q374 643 339 648Q300 648 281 635Q271 628 270 610T268 481V346H284Q327 346 375 352Q421 364 439 392T457 496ZM492 537T492 496T488 427T478 389T469 371T464 361Q464 360 465 360Q469 360 497 370Q593 400 593 495Q593 592 477 630L457 637L461 626Q474 611 488 561Q492 537 492 496ZM464 243Q411 317 410 317Q404 317 401 315Q384 315 370 312H346L526 35H619L606 50Q553 109 464 243Z"></path><path id="MJX-7251-TEX-N-2216" d="M56 731Q56 740 62 745T75 750Q85 750 92 740Q96 733 270 255T444 -231Q444 -239 438 -244T424 -250Q414 -250 407 -240Q404 -236 230 242T56 731Z"></path><path id="MJX-7251-TEX-LO-7B" d="M547 -643L541 -649H528Q515 -649 503 -645Q324 -582 293 -466Q289 -449 289 -428T287 -200L286 42L284 53Q274 98 248 135T196 190T146 222L121 235Q119 239 119 250Q119 262 121 266T133 273Q262 336 284 449L286 460L287 701Q287 737 287 794Q288 949 292 963Q293 966 293 967Q325 1080 508 1148Q516 1150 527 1150H541L547 1144V1130Q547 1117 546 1115T536 1109Q480 1086 437 1046T381 950L379 940L378 699Q378 657 378 594Q377 452 374 438Q373 437 373 436Q350 348 243 282Q192 257 186 254L176 251L188 245Q211 236 234 223T287 189T340 135T373 65Q373 64 374 63Q377 49 378 -93Q378 -156 378 -198L379 -438L381 -449Q393 -504 436 -544T536 -608Q544 -611 545 -613T547 -629V-643Z"></path><path id="MJX-7251-TEX-N-B1" d="M56 320T56 333T70 353H369V502Q369 651 371 655Q376 666 388 666Q402 666 405 654T409 596V500V353H707Q722 345 722 333Q722 320 707 313H409V40H707Q722 32 722 20T707 0H70Q56 7 56 20T70 40H369V313H70Q56 320 56 333Z"></path><path id="MJX-7251-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-7251-TEX-N-221A" d="M95 178Q89 178 81 186T72 200T103 230T169 280T207 309Q209 311 212 311H213Q219 311 227 294T281 177Q300 134 312 108L397 -77Q398 -77 501 136T707 565T814 786Q820 800 834 800Q841 800 846 794T853 782V776L620 293L385 -193Q381 -200 366 -200Q357 -200 354 -197Q352 -195 256 15L160 225L144 214Q129 202 113 190T95 178Z"></path><path id="MJX-7251-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-7251-TEX-LO-7D" d="M119 1130Q119 1144 121 1147T135 1150H139Q151 1150 182 1138T252 1105T326 1046T373 964Q378 942 378 702Q378 469 379 462Q386 394 439 339Q482 296 535 272Q544 268 545 266T547 251Q547 241 547 238T542 231T531 227T510 217T477 194Q390 129 379 39Q378 32 378 -201Q378 -441 373 -463Q342 -580 165 -644Q152 -649 139 -649Q125 -649 122 -646T119 -629Q119 -622 119 -619T121 -614T124 -610T132 -607T143 -602Q195 -579 235 -539T285 -447Q286 -435 287 -199T289 51Q294 74 300 91T329 138T390 197Q412 213 436 226T475 244L489 250L472 258Q455 265 430 279T377 313T327 366T293 434Q289 451 289 472T287 699Q286 941 285 948Q279 978 262 1005T227 1048T184 1080T151 1100T129 1109L127 1110Q119 1113 119 1130Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-7251-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(806.8, 0)"><use xlink:href="#MJX-7251-TEX-N-2208"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(1751.6, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-7251-TEX-D-211D"></use></g></g><g data-mml-node="mi" transform="translate(2473.6, 0)"><use xlink:href="#MJX-7251-TEX-N-2216"></use></g><g data-mml-node="mrow" transform="translate(2973.6, 0)"><g data-mml-node="mo"><use xlink:href="#MJX-7251-TEX-LO-7B"></use></g><g data-mml-node="mo" transform="translate(667, 0)"><use xlink:href="#MJX-7251-TEX-N-B1"></use></g><g data-mml-node="mfrac" transform="translate(1445, 0)"><g data-mml-node="mn" transform="translate(521.6, 394) scale(0.707)"><use xlink:href="#MJX-7251-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220, -519.2) scale(0.707)"><g transform="translate(853, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-7251-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(0, 118.1)"><use xlink:href="#MJX-7251-TEX-N-221A"></use></g><rect width="500" height="42.4" x="853" y="875.6"></rect></g><rect width="1156.7" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(2841.7, 0)"><use xlink:href="#MJX-7251-TEX-LO-7D"></use></g></g></g></g></svg></mjx-container></p>

<p>Eine Matrix $A$ ist invertierbar, falls $\det(A) \neq 0$ ist.</p>


<p>Eine Matrix <mjx-container class="MathJax" jax="SVG"><svg alt="A" style="vertical-align: 0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-7242-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-7242-TEX-I-1D434"></use></g></g></g></svg></mjx-container> ist invertierbar, falls <mjx-container class="MathJax" jax="SVG"><svg alt="\det(A) \neq 0" style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="10.748ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4750.6 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-7243-TEX-N-64" d="M376 495Q376 511 376 535T377 568Q377 613 367 624T316 637H298V660Q298 683 300 683L310 684Q320 685 339 686T376 688Q393 689 413 690T443 693T454 694H457V390Q457 84 458 81Q461 61 472 55T517 46H535V0Q533 0 459 -5T380 -11H373V44L365 37Q307 -11 235 -11Q158 -11 96 50T34 215Q34 315 97 378T244 442Q319 442 376 393V495ZM373 342Q328 405 260 405Q211 405 173 369Q146 341 139 305T131 211Q131 155 138 120T173 59Q203 26 251 26Q322 26 373 103V342Z"></path><path id="MJX-7243-TEX-N-65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z"></path><path id="MJX-7243-TEX-N-74" d="M27 422Q80 426 109 478T141 600V615H181V431H316V385H181V241Q182 116 182 100T189 68Q203 29 238 29Q282 29 292 100Q293 108 293 146V181H333V146V134Q333 57 291 17Q264 -10 221 -10Q187 -10 162 2T124 33T105 68T98 100Q97 107 97 248V385H18V422H27Z"></path><path id="MJX-7243-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-7243-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-7243-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-7243-TEX-N-2260" d="M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z"></path><path id="MJX-7243-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-7243-TEX-N-64"></use><use xlink:href="#MJX-7243-TEX-N-65" transform="translate(556, 0)"></use><use xlink:href="#MJX-7243-TEX-N-74" transform="translate(1000, 0)"></use></g><g data-mml-node="mo" transform="translate(1389, 0)"><use xlink:href="#MJX-7243-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1778, 0)"><use xlink:href="#MJX-7243-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(2528, 0)"><use xlink:href="#MJX-7243-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3194.8, 0)"><use xlink:href="#MJX-7243-TEX-N-2260"></use></g><g data-mml-node="mn" transform="translate(4250.6, 0)"><use xlink:href="#MJX-7243-TEX-N-30"></use></g></g></g></svg></mjx-container> ist.</p>


<p>Berechne zunächst $\det(A)$ mit der Regel von Sarrus:</p>


<p>Berechne zunächst <mjx-container class="MathJax" jax="SVG"><svg alt="\det(A)" style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="6.6ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2917 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-7244-TEX-N-64" d="M376 495Q376 511 376 535T377 568Q377 613 367 624T316 637H298V660Q298 683 300 683L310 684Q320 685 339 686T376 688Q393 689 413 690T443 693T454 694H457V390Q457 84 458 81Q461 61 472 55T517 46H535V0Q533 0 459 -5T380 -11H373V44L365 37Q307 -11 235 -11Q158 -11 96 50T34 215Q34 315 97 378T244 442Q319 442 376 393V495ZM373 342Q328 405 260 405Q211 405 173 369Q146 341 139 305T131 211Q131 155 138 120T173 59Q203 26 251 26Q322 26 373 103V342Z"></path><path id="MJX-7244-TEX-N-65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z"></path><path id="MJX-7244-TEX-N-74" d="M27 422Q80 426 109 478T141 600V615H181V431H316V385H181V241Q182 116 182 100T189 68Q203 29 238 29Q282 29 292 100Q293 108 293 146V181H333V146V134Q333 57 291 17Q264 -10 221 -10Q187 -10 162 2T124 33T105 68T98 100Q97 107 97 248V385H18V422H27Z"></path><path id="MJX-7244-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-7244-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-7244-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-7244-TEX-N-64"></use><use xlink:href="#MJX-7244-TEX-N-65" transform="translate(556, 0)"></use><use xlink:href="#MJX-7244-TEX-N-74" transform="translate(1000, 0)"></use></g><g data-mml-node="mo" transform="translate(1389, 0)"><use xlink:href="#MJX-7244-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1778, 0)"><use xlink:href="#MJX-7244-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(2528, 0)"><use xlink:href="#MJX-7244-TEX-N-29"></use></g></g></g></svg></mjx-container> mit der Regel von Sarrus:</p>


<p>\begin{align*}</p>
<p>\operatorname{det}(A) &amp;= \operatorname{det}\left(\begin{array}{ccc}</p>
<p>1 &amp; -a &amp; 0 \\</p>
<p>-a &amp; 1 &amp; -a \\</p>
<p>0 &amp; -a &amp; 1</p>
<p>\end{array}\right) \\</p>
<p>&amp;= 1+0+0-0-a^{2}-a^{2} \\</p>
<p>&amp;= 1 - 2a^2</p>
<p>\end{align*}</p>


<p>Untersuche nun welche Werte für $a$ die Bedingung $\det(A) \neq 0$ erfüllen:</p>


<p>Untersuche nun welche Werte für <mjx-container class="MathJax" jax="SVG"><svg alt="a" style="vertical-align: -0.023ex" xmlns="http://www.w3.org/2000/svg" width="1.197ex" height="1.02ex" role="img" focusable="false" viewBox="0 -441 529 451" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-7246-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-7246-TEX-I-1D44E"></use></g></g></g></svg></mjx-container> die Bedingung <mjx-container class="MathJax" jax="SVG"><svg alt="\det(A) \neq 0" style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="10.748ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4750.6 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-7247-TEX-N-64" d="M376 495Q376 511 376 535T377 568Q377 613 367 624T316 637H298V660Q298 683 300 683L310 684Q320 685 339 686T376 688Q393 689 413 690T443 693T454 694H457V390Q457 84 458 81Q461 61 472 55T517 46H535V0Q533 0 459 -5T380 -11H373V44L365 37Q307 -11 235 -11Q158 -11 96 50T34 215Q34 315 97 378T244 442Q319 442 376 393V495ZM373 342Q328 405 260 405Q211 405 173 369Q146 341 139 305T131 211Q131 155 138 120T173 59Q203 26 251 26Q322 26 373 103V342Z"></path><path id="MJX-7247-TEX-N-65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z"></path><path id="MJX-7247-TEX-N-74" d="M27 422Q80 426 109 478T141 600V615H181V431H316V385H181V241Q182 116 182 100T189 68Q203 29 238 29Q282 29 292 100Q293 108 293 146V181H333V146V134Q333 57 291 17Q264 -10 221 -10Q187 -10 162 2T124 33T105 68T98 100Q97 107 97 248V385H18V422H27Z"></path><path id="MJX-7247-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-7247-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-7247-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-7247-TEX-N-2260" d="M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z"></path><path id="MJX-7247-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mo"><use xlink:href="#MJX-7247-TEX-N-64"></use><use xlink:href="#MJX-7247-TEX-N-65" transform="translate(556, 0)"></use><use xlink:href="#MJX-7247-TEX-N-74" transform="translate(1000, 0)"></use></g><g data-mml-node="mo" transform="translate(1389, 0)"><use xlink:href="#MJX-7247-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1778, 0)"><use xlink:href="#MJX-7247-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(2528, 0)"><use xlink:href="#MJX-7247-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3194.8, 0)"><use xlink:href="#MJX-7247-TEX-N-2260"></use></g><g data-mml-node="mn" transform="translate(4250.6, 0)"><use xlink:href="#MJX-7247-TEX-N-30"></use></g></g></g></svg></mjx-container> erfüllen:</p>


<p>\begin{align*}</p>
<p>&amp;&amp; \det(A) &amp;\neq 0 \\</p>
<p>\Leftrightarrow &amp;&amp; a^{2} &amp;\neq \frac{1}{2} \\</p>
<p>\Leftrightarrow &amp;&amp; a &amp;\neq \pm \frac{1}{\sqrt{2}}</p>
<p>\end{align*}</p>

<p>Untersuche für welche Werte von $a$ eine Inverse $A^{-1}$ existiert.</p>
<p></p>
<p>\begin{align*}A=\left(\begin{array}{ccc}</p>
<p>1 &amp; -a &amp; 0 \\</p>
<p>-a &amp; 1 &amp; -a \\</p>
<p>0 &amp; -a &amp; 1</p>
<p>\end{array}\right)\end{align*}</p>

<p>Untersuche für welche Werte von <mjx-container class="MathJax" jax="SVG"><svg alt="a" style="vertical-align: -0.023ex" xmlns="http://www.w3.org/2000/svg" width="1.197ex" height="1.02ex" role="img" focusable="false" viewBox="0 -441 529 451" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-7239-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-7239-TEX-I-1D44E"></use></g></g></g></svg></mjx-container> eine Inverse <mjx-container class="MathJax" jax="SVG"><svg alt="A^{-1}" style="vertical-align: 0" xmlns="http://www.w3.org/2000/svg" width="3.854ex" height="1.887ex" role="img" focusable="false" viewBox="0 -833.9 1703.7 833.9" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-7240-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-7240-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-7240-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use xlink:href="#MJX-7240-TEX-I-1D434"></use></g><g data-mml-node="TeXAtom" transform="translate(750, 363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use xlink:href="#MJX-7240-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778, 0)"><use xlink:href="#MJX-7240-TEX-N-31"></use></g></g></g></g></g></svg></mjx-container> existiert.</p>
<p></p>
<p><mjx-container class="MathJax" jax="SVG"><svg alt="\begin{align*}A=\left(\begin{array}{ccc}
1 & -a & 0 \\
-a & 1 & -a \\
0 & -a & 1
\end{array}\right)\end{align*}" style="vertical-align: -3.733ex" xmlns="http://www.w3.org/2000/svg" width="22.069ex" height="8.597ex" role="img" focusable="false" viewBox="0 -2150 9754.6 3800" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-7241-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-7241-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 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Dies ist eine interaktive Aufgabe zu: Regel von Sarrus (3x3) mit praktischen Tipps zum Lösen und einer Zusammenfassung der nötigen Theorie

Determinanten bestimmen - Regel von Sarrus (3x3) | Aufgabe mit Lösung


<h2 class="content_sub_heading">Determinate einer 3x3 Matrix über Sarrus</h2>
<ol>
<li>Schreibe die ersten beiden Spalten der Matrix noch einmal rechts neben die Matrix.<br />
<p>\begin{align*}</p>
<p>\\</p>
<p>\quad A:=\left(\begin{array}{lll}</p>
<p>a &amp; b &amp; c \\</p>
<p>d &amp; e &amp; f \\</p>
<p>g &amp; h &amp; i</p>
<p>\end{array}\right)</p>
<p>\Rightarrow</p>
<p>\begin{array}{|lll|ll}</p>
<p>\color{#5FAFD2}{a} &amp;\color{#AAC869}{ b} &amp;\color{#DC6955}{c} &amp;a&amp;b\\</p>
<p>d &amp;\color{#5FAFD2}{ e} &amp;\color{#AAC869}{ f} &amp;\color{#DC6955}{d}&amp;e\\</p>
<p>g &amp; h &amp; \color{#5FAFD2}{i}&amp;\color{#AAC869}{g}&amp;\color{#DC6955}{h}</p>
<p>\end{array} \qquad \qquad</p>
<p>\begin{array}{|lll|ll}</p>
<p>a &amp;b &amp;\color{#5FAFD2}{c} &amp; \color{#AAC869}{a}&amp;\color{#DC6955}{b}\\</p>
<p>d &amp;\color{#5FAFD2}{e} &amp;\color{#AAC869}{f} &amp;\color{#DC6955}{d}&amp;e\\</p>
<p>\color{#5FAFD2}{g} &amp; \color{#AAC869}{h} &amp; \color{#DC6955}{i}&amp;g&amp;h</p>
<p>\end{array}</p>
<p>\end{align*}<br /><br /></p>
</li>
<li>
<p>Addiere die Produkte der entstandenen drei Diagonalen von links oben nach rechts unten und subtrahiere dann die Produkte der Diagonalen von rechts oben nach links unten:</p>
<p>\begin{align*}</p>
<p>\quad \Rightarrow\det(A) =\color{#5FAFD2}{a e i} \ \color{#000}{+} \ \color{#AAC869}{b f g} \ \color{#000}{+} \ \color{#DC6955}{c d h} \ \color{#000}{-} \ \color{#DC6955}{b d i} \ \color{#000}{-} \ \color{#AAC869}{a f h} \ \color{#000}{-} \ \color{#5FAFD2}{c e g}</p>
<p>\end{align*}</p>
</li>
</ol>


Aufgabenstellung:

Lösungsweg:

Lösung: