Inverses of Binary 3x3 matrices

A 3x3 boolean matrix may be expressed as 3 octal digits, each row corresponding to a digit. The identity matrix is 421, and here are the 168 possible invertible matrices: 124*124 = 125*524 = 126*324 = 127*724 = 134*164 = 135*564 = 136*764 = 137*364 = 142*214 = 143*254 = 146*234 = 147*274 = 152*614 = 153*654 = 156*674 = 157*634 = 162*314 = 163*754 = 164*134 = 165*574 = 172*714 = 173*354 = 174*174 = 175*534 = 214*142 = 215*342 = 216*542 = 217*742 = 234*146 = 235*746 = 236*546 = 237*346 = 241*241 = 243*245 = 245*243 = 247*247 = 251*341 = 253*745 = 254*143 = 256*547 = 261*641 = 263*645 = 265*647 = 267*643 = 271*741 = 273*345 = 274*147 = 276*543 = 314*162 = 315*362 = 316*762 = 317*562 = 324*126 = 325*726 = 326*326 = 327*526 = 341*251 = 342*215 = 345*273 = 346*237 = 351*351 = 352*715 = 354*173 = 357*537 = 361*751 = 362*315 = 364*137 = 367*573 = 371*651 = 372*615 = 375*637 = 376*673 = 412*412 = 413*432 = 416*452 = 417*472 = 421*421 = 423*423 = 425*425 = 427*427 = 431*431 = 432*413 = 435*475 = 436*457 = 452*416 = 453*476 = 456*456 = 457*436 = 461*461 = 463*467 = 465*465 = 467*463 = 471*471 = 472*417 = 475*435 = 476*453 = 512*612 = 513*632 = 516*672 = 517*652 = 521*521 = 523*723 = 524*125 = 526*327 = 531*531 = 532*713 = 534*175 = 537*357 = 542*216 = 543*276 = 546*236 = 547*256 = 561*571 = 562*317 = 564*135 = 567*753 = 571*561 = 573*367 = 574*165 = 576*763 = 612*512 = 613*732 = 614*152 = 615*372 = 621*621 = 623*623 = 625*627 = 627*625 = 631*731 = 632*513 = 634*157 = 637*375 = 641*261 = 643*267 = 645*263 = 647*265 = 651*371 = 652*517 = 654*153 = 657*735 = 672*516 = 673*376 = 674*156 = 675*736 = 712*712 = 713*532 = 714*172 = 715*352 = 721*721 = 723*523 = 724*127 = 726*325 = 731*631 = 732*613 = 735*657 = 736*675 = 741*271 = 742*217 = 745*253 = 746*235 = 751*361 = 753*567 = 754*163 = 756*765 = 762*316 = 763*576 = 764*136 = 765*756 = 421

The 22 square roots of unity (identity) are 124² = 174² = 241² = 247² = 326² = 351² = 412² = 421² = 423² = 425² = 427² = 431² = 456² = 461² = 465² = 471² = 521² = 531² = 621² = 623² = 712² = 721² = 421

There are also 22 square roots of zero: 000² = 002² = 004² = 006² = 010² = 033² = 040² = 044² = 050² = 077² = 100² = 110² = 200² = 202² = 300² = 333² = 505² = 555² = 660² = 666² = 707² = 770² = 000

There 260 squares: 000 001 002 003 004 005 006 007 010 011 013 017 020 021 022 025 030 032 033 036 040 044 050 053 055 057 060 061 065 066 070 072 076 077 100 101 105 107 110 111 113 115 121 124 125 127 132 135 136 137 142 143 145 146 152 155 156 157 162 163 165 166 171 173 174 175 200 202 213 214 215 216 220 221 222 227 232 234 235 237 241 247 253 254 255 256 260 261 262 263 264 265 266 267 270 271 273 276 300 303 305 307 313 314 316 317 321 325 326 327 330 332 333 334 342 343 345 346 351 352 354 356 360 361 364 367 370 371 372 375 400 401 403 404 411 412 413 417 420 421 422 423 424 425 426 427 430 431 432 436 440 441 444 447 451 453 456 457 461 465 471 472 474 476 500 504 505 506 513 514 516 517 520 521 524 526 530 531 532 534 542 543 544 547 550 552 554 555 562 563 564 567 573 574 576 577 600 601 603 606 613 614 615 616 621 623 632 634 635 637 640 641 642 643 644 645 646 647 650 651 652 657 660 666 672 674 675 677 700 704 706 707 711 712 713 715 721 722 724 726 732 735 736 737 740 741 745 746 750 751 753 754 762 763 764 767 770 773 775 777 .