有多少个2*2的矩阵满足

nyy

矩阵(2,3;6,9)的平方=(22,33;66,99)
逛知乎发现的！
有多少个矩阵这样？

lsr314

{{2,2},{9,9}}
{{2,3},{6,9}}
{{2,6},{3,9}}
{{2,9},{2,9}}
{{3,3},{8,8}}
{{3,4},{6,8}}
{{3,6},{4,8}}
{{3,8},{3,8}}
{{4,4},{7,7}}
{{4,7},{4,7}}
{{5,5},{6,6}}
{{5,6},{5,6}}
以及它们的转置矩阵

mathe

[2,3;6,9]
[3,4;6,8]
[10,13;70,91]
[10,14;65,91]
[10,26;35,91]
[11,15;66,90]
[11,18;55,90]
[11,22;45,90]
[11,30;33,90]
[13,22;52,88]
[13,26;44,88]
[14,21;58,87]
[14,29;42,87]
[15,30;43,86]
[16,17;80,85]
[16,20;68,85]
[16,34;40,85]
[17,21;68,84]
[17,28;51,84]
[17,34;42,84]
[19,38;41,82]
[20,27;60,81]
[20,30;54,81]
[20,36;45,81]
[21,24;70,80]
[21,28;60,80]
[21,30;56,80]
[21,35;48,80]
[21,40;42,80]
[23,26;69,78]
[23,39;46,78]
[24,28;66,77]
[24,33;56,77]
[24,42;44,77]
[25,38;50,76]
[26,30;65,75]
[26,39;50,75]
[27,37;54,74]
[29,36;58,72]
[31,35;62,70]
[32,46;48,69]
[33,34;66,68]
[33,44;51,68]
[35,42;55,66]
[36,39;60,65]
[36,45;52,65]
[38,42;57,63]
[100,106;850,901]
[100,170;530,901]
[100,212;425,901]
[100,265;340,901]
[101,150;606,900]
[101,180;505,900]
[101,202;450,900]
[101,225;404,900]
[101,300;303,900]
[102,174;527,899]
[102,186;493,899]
[103,206;449,898]
[104,138;676,897]
[104,156;598,897]
[104,169;552,897]
[104,184;507,897]
[104,276;338,897]
[104,299;312,897]
[105,112;840,896]
[105,120;784,896]
[105,128;735,896]
[105,140;672,896]
[105,147;640,896]
[105,160;588,896]
[105,168;560,896]
[105,192;490,896]
[105,196;480,896]
[105,210;448,896]
[105,224;420,896]
[105,240;392,896]
[105,245;384,896]
[105,280;336,896]
[105,294;320,896]
[106,179;530,895]
[106,265;358,895]
[107,149;642,894]
[107,214;447,894]
[107,298;321,894]
[108,114;846,893]
[108,141;684,893]
[108,171;564,893]
[108,188;513,893]
[108,228;423,893]
[108,282;342,893]
[109,218;446,892]
[109,223;436,892]
[110,121;810,891]
[110,135;726,891]
[110,162;605,891]
[110,165;594,891]
[110,198;495,891]
[110,242;405,891]
[110,270;363,891]
[110,297;330,891]
[111,178;555,890]
[111,185;534,890]
[111,222;445,890]
[111,267;370,890]
[112,127;784,889]
[112,196;508,889]
[112,254;392,889]
[113,148;678,888]
[113,222;452,888]
[113,226;444,888]
[113,296;339,888]
[115,230;443,886]
[116,118;870,885]
[116,145;708,885]
[116,174;590,885]
[116,177;580,885]
[116,236;435,885]
[116,290;354,885]
[116,295;348,885]
[117,153;676,884]
[117,156;663,884]
[117,169;612,884]
[117,204;507,884]
[117,221;468,884]
[117,234;442,884]
[117,306;338,884]
[119,126;833,882]
[119,147;714,882]
[119,153;686,882]
[119,238;441,882]
[119,294;357,882]
[119,306;343,882]
[121,176;605,880]
[121,220;484,880]
[121,242;440,880]
[122,183;586,879]
[122,293;366,879]
[123,246;439,878]
[125,146;750,876]
[125,150;730,876]
[125,219;500,876]
[125,250;438,876]
[125,292;375,876]
[125,300;365,876]
[126,147;750,875]
[126,150;735,875]
[126,175;630,875]
[126,210;525,875]
[126,225;490,875]
[126,245;450,875]
[126,250;441,875]
[126,294;375,875]
[126,315;350,875]
[127,254;437,874]
[128,144;776,873]
[128,192;582,873]
[128,194;576,873]
[128,288;388,873]
[128,291;384,873]
[129,172;654,872]
[129,218;516,872]
[129,258;436,872]
[129,327;344,872]
[130,134;845,871]
[130,169;670,871]
[130,335;338,871]
[131,145;786,870]
[131,174;655,870]
[131,262;435,870]
[131,290;393,870]
[132,158;726,869]
[132,237;484,869]
[132,242;474,869]
[132,316;363,869]
[133,196;589,868]
[133,217;532,868]
[133,266;434,868]
[134,201;578,867]
[134,289;402,867]
[135,270;433,866]
[136,170;692,865]
[136,173;680,865]
[136,340;346,865]
[137,144;822,864]
[137,216;548,864]
[137,274;432,864]
[137,288;411,864]
[139,278;431,862]
[140,147;820,861]
[140,164;735,861]
[140,196;615,861]
[140,205;588,861]
[140,210;574,861]
[140,245;492,861]
[140,246;490,861]
[140,287;420,861]
[140,294;410,861]
[141,172;705,860]
[141,188;645,860]
[141,215;564,860]
[141,235;516,860]
[141,258;470,860]
[141,282;430,860]
[143,169;726,858]
[143,242;507,858]
[143,286;429,858]
[143,338;363,858]
[145,214;580,856]
[145,232;535,856]
[145,290;428,856]
[146,171;730,855]
[146,190;657,855]
[146,219;570,855]
[146,285;438,855]
[146,342;365,855]
[147,183;686,854]
[147,294;427,854]
[147,343;366,854]
[149,213;596,852]
[149,284;447,852]
[149,298;426,852]
[150,185;690,851]
[150,222;575,851]
[150,230;555,851]
[150,345;370,851]
[151,170;755,850]
[151,302;425,850]
[152,228;566,849]
[152,283;456,849]
[153,159;816,848]
[153,204;636,848]
[153,212;612,848]
[153,272;477,848]
[153,306;424,848]
[153,318;408,848]
[154,242;539,847]
[155,186;705,846]
[155,235;558,846]
[155,279;470,846]
[155,282;465,846]
[155,310;423,846]
[156,169;780,845]
[156,195;676,845]
[156,260;507,845]
[156,338;390,845]
[157,211;628,844]
[157,314;422,844]
[158,237;562,843]
[158,281;474,843]
[159,318;421,842]
[160,232;580,841]
[160,290;464,841]
[161,168;805,840]
[161,184;735,840]
[161,196;690,840]
[161,210;644,840]
[161,230;588,840]
[161,245;552,840]
[161,276;490,840]
[161,280;483,840]
[161,294;460,840]
[161,322;420,840]
[161,345;392,840]
[163,326;419,838]
[164,186;738,837]
[164,246;558,837]
[164,279;492,837]
[164,369;372,837]
[165,190;726,836]
[165,209;660,836]
[165,220;627,836]
[165,228;605,836]
[165,242;570,836]
[165,285;484,836]
[165,330;418,836]
[165,363;380,836]
[166,167;830,835]
[166,334;415,835]
[167,278;501,834]
[167,334;417,834]
[168,196;714,833]
[168,204;686,833]
[168,238;588,833]
[168,294;476,833]
[168,343;408,833]
[168,357;392,833]
[169,208;676,832]
[169,338;416,832]
[170,255;554,831]
[170,277;510,831]
[171,190;747,830]
[171,249;570,830]
[171,285;498,830]
[171,342;415,830]
[173,207;692,828]
[173,276;519,828]
[173,346;414,828]
[175,245;590,826]
[175,295;490,826]
[175,350;413,826]
[176,200;726,825]
[176,220;660,825]
[176,240;605,825]
[176,242;600,825]
[176,264;550,825]
[176,275;528,825]
[176,300;484,825]
[176,330;440,825]
[176,363;400,825]
[177,206;708,824]
[177,236;618,824]
[177,309;472,824]
[177,354;412,824]
[179,274;537,822]
[179,358;411,822]
[181,205;724,820]
[181,362;410,820]
[182,234;637,819]
[182,273;546,819]
[182,294;507,819]
[182,338;441,819]
[183,366;409,818]
[184,344;437,817]
[185,204;740,816]
[185,222;680,816]
[185,240;629,816]
[185,255;592,816]
[185,272;555,816]
[185,296;510,816]
[185,340;444,816]
[185,370;408,816]
[186,310;489,815]
[186,326;465,815]
[187,242;629,814]
[187,374;407,814]
[188,271;564,813]
[188,282;542,813]
[189,196;783,812]
[189,203;756,812]
[189,252;609,812]
[189,261;588,812]
[189,294;522,812]
[189,348;441,812]
[189,378;406,812]
[191,270;573,810]
[191,382;405,810]
[193,202;772,808]
[193,386;404,808]
[194,269;582,807]
[194,291;538,807]
[195,310;507,806]
[195,338;465,806]
[195,390;403,806]
[196,230;686,805]
[196,245;644,805]
[196,322;490,805]
[196,343;460,805]
[197,201;788,804]
[197,268;591,804]
[197,394;402,804]
[198,219;726,803]
[198,242;657,803]
[198,363;438,803]
[199,398;401,802]
[200,225;712,801]
[200,267;600,801]
[200,300;534,801]
[200,356;450,801]
[200,360;445,801]
[201,240;670,800]
[201,268;600,800]
[201,300;536,800]
[201,335;480,800]
[201,400;402,800]
[203,266;609,798]
[203,294;551,798]
[203,399;406,798]
[205,398;410,796]
[206,265;618,795]
[206,309;530,795]
[206,318;515,795]
[207,397;414,794]
[208,244;676,793]
[208,338;488,793]
[209,228;726,792]
[209,242;684,792]
[209,264;627,792]
[209,342;484,792]
[209,363;456,792]
[209,396;418,792]
[210,226;735,791]
[210,245;678,791]
[210,294;565,791]
[210,339;490,791]
[211,395;422,790]
[212,263;636,789]
[212,318;526,789]
[213,284;591,788]
[213,394;426,788]
[215,258;655,786]
[215,262;645,786]
[215,393;430,786]
[216,270;628,785]
[216,314;540,785]
[216,360;471,785]
[217,248;686,784]
[217,343;496,784]
[217,392;434,784]
[218,261;654,783]
[218,327;522,783]
[219,391;438,782]
[220,242;710,781]
[220,284;605,781]
[220,355;484,781]
[221,255;676,780]
[221,260;663,780]
[221,338;510,780]
[221,340;507,780]
[221,390;442,780]
[222,246;703,779]
[223,389;446,778]
[224,259;672,777]
[224,294;592,777]
[224,296;588,777]
[224,336;518,777]
[224,392;444,777]
[225,291;600,776]
[225,300;582,776]
[225,360;485,776]
[225,388;450,776]
[226,310;565,775]
[227,258;681,774]
[227,387;454,774]
[229,386;458,772]
[230,257;690,771]
[230,345;514,771]
[231,242;735,770]
[231,245;726,770]
[231,294;605,770]
[231,330;539,770]
[231,363;490,770]
[231,385;462,770]
[233,256;699,768]
[233,384;466,768]
[234,338;531,767]
[234,354;507,767]
[235,383;470,766]
[236,255;708,765]
[236,295;612,765]
[236,306;590,765]
[236,340;531,765]
[236,354;510,765]
[237,316;573,764]
[237,382;474,764]
[239,254;717,762]
[239,381;478,762]
[241,380;482,760]
[242,253;726,759]
[242,363;506,759]
[243,379;486,758]
[245,252;735,756]
[245,270;686,756]
[245,294;630,756]
[245,315;588,756]
[245,343;540,756]
[245,378;490,756]
[245,420;441,756]
[246,302;615,755]
[246,410;453,755]
[247,338;551,754]
[247,377;494,754]
[248,251;744,753]
[248,372;502,753]
[249,282;664,752]
[249,332;564,752]
[249,376;498,752]
[251,375;502,750]
[252,294;642,749]
[252,321;588,749]
[252,428;441,749]
[253,374;506,748]
[253,391;484,748]
[254,381;498,747]
[255,373;510,746]
[256,298;640,745]
[256,320;596,745]
[257,372;514,744]
[259,371;518,742]
[260,285;676,741]
[260,338;570,741]
[260,380;507,741]
[260,390;494,741]
[261,290;666,740]
[261,333;580,740]
[261,348;555,740]
[261,370;522,740]
[261,435;444,740]
[263,369;526,738]
[264,268;726,737]
[264,363;536,737]
[264,402;484,737]
[265,368;530,736]
[265,424;460,736]
[266,285;686,735]
[266,294;665,735]
[266,343;570,735]
[266,399;490,735]
[267,367;534,734]
[269,366;538,732]
[270,306;645,731]
[270,387;510,731]
[270,430;459,731]
[271,365;542,730]
[272,306;648,729]
[272,324;612,729]
[272,408;486,729]
[272,432;459,729]
[273,294;676,728]
[273,312;637,728]
[273,338;588,728]
[273,364;546,728]
[273,392;507,728]
[275,330;605,726]
[275,363;550,726]
[276,290;690,725]
[276,300;667,725]
[276,345;580,725]
[276,348;575,725]
[276,435;460,725]
[277,362;554,724]
[278,417;482,723]
[279,342;589,722]
[279,361;558,722]
[280,392;515,721]
[280,412;490,721]
[281,360;562,720]
[283,359;566,718]
[284,426;478,717]
[285,358;570,716]
[285,380;537,716]
[286,338;605,715]
[287,294;697,714]
[287,357;574,714]
[288,368;558,713]
[288,372;552,713]
[288,414;496,713]
[289,356;578,712]
[290,395;522,711]
[290,435;474,711]
[291,355;582,710]
[291,426;485,710]
[293,354;586,708]
[294,303;686,707]
[294,343;606,707]
[295,353;590,706]
[296,370;564,705]
[296,376;555,705]
[296,444;470,705]
[297,352;594,704]
[297,363;576,704]
[297,396;528,704]
[297,432;484,704]
[299,338;621,702]
[299,351;598,702]
[299,414;507,702]
[301,350;602,700]
[301,430;490,700]
[302,453;466,699]
[303,349;606,698]
[304,323;656,697]
[304,328;646,697]
[305,348;610,696]
[305,366;580,696]
[305,435;488,696]
[306,417;510,695]
[307,347;614,694]
[308,363;588,693]
[308,396;539,693]
[308,441;484,693]
[309,346;618,692]
[309,412;519,692]
[311,345;622,690]
[312,318;676,689]
[312,338;636,689]
[312,424;507,689]
[313,344;626,688]
[314,458;471,687]
[315,343;630,686]
[315,441;490,686]
[316,395;548,685]
[317,342;634,684]
[319,341;638,682]
[320,454;480,681]
[321,340;642,680]
[321,408;535,680]
[321,428;510,680]
[323,339;646,678]
[325,338;650,676]
[326,450;489,675]
[327,337;654,674]
[329,336;658,672]
[329,376;588,672]
[329,392;564,672]
[330,363;610,671]
[330,366;605,671]
[331,335;662,670]
[332,446;498,669]
[333,334;666,668]
[333,444;501,668]
[335,370;603,666]
[335,402;555,666]
[336,380;588,665]
[336,392;570,665]
[336,399;560,665]
[336,420;532,665]
[336,456;490,665]
[338,442;507,663]
[341,363;620,660]
[341,372;605,660]
[341,465;484,660]
[344,387;584,657]
[344,438;516,657]
[345,368;615,656]
[345,410;552,656]
[345,460;492,656]
[350,434;525,651]
[350,465;490,651]
[351,390;585,650]
[351,450;507,650]
[352,472;484,649]
[356,430;534,645]
[356,445;516,645]
[357,391;588,644]
[357,476;483,644]
[361,380;608,640]
[362,426;543,639]
[365,438;530,636]
[366,381;610,635]
[368,422;552,633]
[369,474;492,632]
[371,441;530,630]
[371,477;490,630]
[372,444;527,629]
[374,418;561,627]
[376,470;500,625]
[377,464;507,624]
[378,441;534,623]
[380,414;570,621]
[380,437;540,621]
[380,460;513,621]
[381,465;508,620]
[385,392;605,616]
[385,440;539,616]
[385,484;490,616]
[386,410;579,615]
[390,470;507,611]
[392,406;588,609]
[393,456;524,608]
[395,474;505,606]
[396,484;495,605]
[398,402;597,603]
[405,447;540,596]
[406,490;493,595]
[416,468;520,585]
[416,480;507,585]
[417,438;556,584]
[420,490;498,581]
[425,450;544,576]
[425,480;510,576]
[429,484;507,572]
[434,441;558,567]
[436,452;545,565]
[440,484;510,561]
[441,490;504,560]
[450,475;522,551]
[455,490;507,546]
[468,492;507,533]
[473,484;516,528]
[476,490;510,525]

mathe

可以看出如果对于矩阵
\(\begin{bmatrix}a&b\\c&d\end{bmatrix}\)
满足条件，现在看到情况它们的位数都相同，而容易轻松证明b和c位数必须相同。
而如果它们全部位数相同，为k位数，设\(K=10^k+1\)
那么必然有a+d=K, bc=ad,所以可以将K拆分成两个不同k位数a,d之和，而且要求它们的乘积还可以表示为另外两个不同k位数b,c的乘积。