证法一：设[tex=4.143x1.0]uqh+oOvD2P9iqZ7dD7XO1L6SBweiS/Qz8LkmpATeeLY=[/tex]，则存在[tex=0.929x0.786]VF0GLe2VBE/4VKNzpyOfFg==[/tex]阶和[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶可逆矩阵[tex=0.786x1.286]dSWbQCTjdbLxKy7q0ps2gg==[/tex]和[tex=0.857x1.214]to/MrMoO1ux8UhZHnpEvBg==[/tex]，使得[tex=8.571x2.786]GOgGvXf8fpWrP7XGIdsj88kak28o9iG99G9ZZzLGg100Tl7qQiymcM0gD2WuAA/91akGPX+gzufDSfJ5sxd6Pa8iQhaznaxeokF5BLZ9oW0=[/tex]。必要性。设矩阵方程[tex=3.143x1.0]TfRL0Lu8PApzv7GgboFLtw==[/tex]有解[tex=1.214x1.214]GP4V2CVtVaqD85GNObLHOQ==[/tex]，记[tex=6.857x2.786]a0fLp1xm4hk1VusUhnk9I/3UzfjFk0wCGeZNFVzEB5svGLQTH13wS3ZKUWnz878V4q1AcpuaqfWsQo8MPw6WPWHQkfLmYvYdgf5p1LoyORA=[/tex]，其中 [tex=1.5x1.214]6Cm7a8sfmFUUemNm42fPjw==[/tex]为[tex=2.214x1.143]sOzx+UKgfzKHeX7t5rLfLQ==[/tex]阶矩阵，则[tex=29.143x2.786]RfUv+n6Uq/6qwveaJRrv8dAlQhEAKzvDFfueuNziGNjhEOqWdFbZtynugwuFS9+s/JB/SyzkmIX8R4dAjmWcId8+TQqFdq9/5PfVS8XT8dC7/uaPGGTyb2AlD1Iu8clcOQ/x5TVOjIBxpgvV7by2KRmU/TA7g83V+u6+XQEtiapw0MUwb/WFT79RUheccg70RkyM0Ie970VehQv30rHV6k7ngA5D/M+fFZhwMqa0f2GDpmK2qydqYbRuFRqXYrJ9qf9DqgZJA48wzErcw6McU7dVGTeylvq2Cn1BtuW7fzM6cMMeE13ROUD2HWVL+3aazUdwpRWX5hMDzt4NjSHSQD69mgHoWHd0s57csWSSRkc=[/tex]，而[tex=21.714x2.786]k8PvTJe4iQVkvPfhUhxDGGZc3xxLnajOUdTZn3dS2hHs9AiLdhliPaBEDbLvj35Zno0cI8w17dCwcV66Is6mUiFVXz5mZAr8Z1+KUSq1sdc1H+G1xCo8hKzERYXWpibGt940t7WMnwAduTgMEpnAwrlMBE1fSLlkdx288M+Mzhw9CkK+GICgtU2QQkY5ekTLSCqtaduDnPjJb2Tihk7/c2FsNF6Sh4TmHJDfCh5HimjLmhLkGxApkXdkqrdz5VD0[/tex][tex=16.571x2.786]QCnSjjZlkUmH/xR95KIndvSkRddcPDZjCC/bXyDdfu21FXpaxwSe49BTEe2h+clFokd+GbRB1g0bV43hamOFTaBd4RyrLmm6+t5K/Xvifse9lQq1sSoyc+TipCb+MTooHG2TNdqiHw+iLd3hC88dphz07lCoAPdtlsg4WfCmTONdcn3wtxu39LcJxyvZemWjfWbCImuTVVO1iTBeDvljNQ==[/tex]因为[tex=0.786x1.286]dSWbQCTjdbLxKy7q0ps2gg==[/tex]，[tex=0.786x1.286]gvyykdQdNBydRqWi9I4iuA==[/tex]可逆，我们有[tex=20.929x2.786]k8PvTJe4iQVkvPfhUhxDGGZc3xxLnajOUdTZn3dS2hHs9AiLdhliPaBEDbLvj35ZmZZt6mxbOtS5vNb1qfGnKYsOU0W63IuK+rPxEpgod5lgEO4RRAfgVSflOC/Qryw8FsmAaMJeDXgGQdnrEzBfETJSm/5n/DVn0Z/jgWze7jU+GBtA+Utmv9U0l3y6ZiUsGCOr7YUADSVVs7cGG8qPbA==[/tex]。充分性。设[tex=8.714x1.357]uqh+oOvD2P9iqZ7dD7XO1Au5GpPM3U/ETOAOUQD4vNGJIlG2rC6N74CkHadtFKQAQ+HX1Go/tlXoPtTa4boe2w==[/tex]，由于[tex=25.357x2.786]KjSuBwVVLOgsPqqUeTw3oL9r4/AhGAHu0q1PSX+DlcKAYfhQdsF1glj22aFY2LK/mlk6upk9yFlQ5nN6DD6CUsFD0qNqCaE5YC1Dqdr4uvKgnpDziBgK2aGwOKA2tav1geXsMFDlPNukcs2pkl+KgCmpa8XBu8sc8AYWlm+rZrDWO1ogQLWjouAQlI8sL4kTHeaphc8SDT8GGL6CFk9kdyUoaeMHE/zjSbL8xiDxzNwnQOB1plc2pRM4DuCsv/aa[/tex][tex=30.357x3.071]mo1p6GNFKmZVkJ469bfo7yBgyntv5m21vNALASfJwtCz4teezqLNa9C+Nuc/Ht+79GL+odB1Gl39md552G2GTLAnUrnWN6HENI77n/nXnm7NNniYsTJzHOX+CHbO8/JHPnEqIdberAd20qQhCU//fGJZQKWu+K7oJxlNFU6ieNFP3BctMmWS9nWThihsGiK1b54lRv9Qmgih+dtNn4uD15f1s9RrJ7uwY6aKK3hSQtBr2GQhw4NCVlr2kVxMVv8Gxp6ERpyHHbCzkKgzCpahPTbkm3ea90BYpml9fgqF0FVC0bqJ63rGxLtdT/AMQ/VCafheR3LSHdGI/SATF9jvI8+7s87UYuCYNi6pdIk80x4Qode7WAJ6foqtwQJYCKar[/tex]其中[tex=7.357x2.786]GTBpLQJpZn99kyGvSCjqJYgD2X+qmwHW7hnxSu65hnDoQytWhxP++5TfRxEvG2wU/I2X3ppg9GqzU+GjHkKPUL9Vrdwxmfi9hIkPYaQZWAM=[/tex]，[tex=1.5x1.214]r/qtxDM9YuRtFxaogyCYJg==[/tex]为[tex=2.214x1.143]sOzx+UKgfzKHeX7t5rLfLQ==[/tex]矩阵，所以[tex=20.929x3.071]k8PvTJe4iQVkvPfhUhxDGGZc3xxLnajOUdTZn3dS2hHs9AiLdhliPaBEDbLvj35ZmZZt6mxbOtS5vNb1qfGnKYsOU0W63IuK+rPxEpgod5lgEO4RRAfgVSflOC/Qryw8vVG2vzbqUH6SlZlERHOCZove+8lChKd6LSjAYXFMWo/bSn3MmZCXS2Iat4oy4zZnh+0FAaOS4g3tbcuPe6dBnA==[/tex]。因此[tex=2.786x1.214]gCPt4hpmmoNEKeVCBVO4zw==[/tex]，取[tex=13.357x2.929]Bse7vaTOJHoawUae7QLOEtu1WqAXQmWHFgkp1W4ZmXnf/3OyYuOpNH6RPYmEw+eos1pcQm7K0t1wcJrQaIbq/uazlVcgjIiI9l3VaRXBX/YDCBWiASs/nKdOgzwvnAiJvwCiEER/176zT5RBeZGf+g==[/tex]，则[tex=30.929x2.929]POYaeaigbbUfKKnJkLlWSaojyPNk2iDgEIB4smqLdqqpk2M0gVOcqxNK0gFuYAXDkAaLjgMHAlGJKMLkfV9K4xr9rqVO9m4uBJr8f3xceBRr+aAiPWpzopXLU5GGY2laUW4ZbXsG17U3zYxL/BIulpZuApZL5Rn05NOe8PJIkyUtaJMkFUfUkSn0yphdJ8imScxtubESVLztGdOkEL67JnXRozRIvbMH84enjBO4ewWWuOHRrA5xMp09SHGlhGbkudx2ZfjtYumTvhrmBXHundfQRnfx8Ew0YWELbRTcxsSS8Lz2q0hEmVhjtwXpQszCdnRYXYsixMhLG9hC7W4J0A==[/tex][tex=22.286x2.786]jaqkKt8zEwr29/cypwF+r93jGwOa4DPvqKAguit7xBmP75e8NMJJktMkeqe3sK3nAyr+UaPd6H/nDphNkwjt1rQc7f1mBsDeEnh2aNte9xWXCLki0jIt+MjSSNy2/ja8t0V70EKoB5mLvzs13RfNyLADHZRDYPlvXxGEexWUAacdIl/l2Yw58rojsIg/NEODgqJuYmRdjjuLi9tYzXuXhTvXptP+eFsOmYC7B9aA52o1BNOjPzlPnunYCmQjqhMh[/tex]所以[tex=13.357x2.929]Bse7vaTOJHoawUae7QLOEtu1WqAXQmWHFgkp1W4ZmXnf/3OyYuOpNH6RPYmEw+eos1pcQm7K0t1wcJrQaIbq/uazlVcgjIiI9l3VaRXBX/YDCBWiASs/nKdOgzwvnAiJvwCiEER/176zT5RBeZGf+g==[/tex]是矩阵方程[tex=3.143x1.0]TfRL0Lu8PApzv7GgboFLtw==[/tex]的解。证法二：记[tex=7.929x1.357]RXYLWN+dl4oZYM4ZwbSUQ2RtWVqgCJjDnPr09uxuABDW1r2+s33Qzov9nDSQlBFWyMenbiKfAOmI5qJ1Z6B1OA==[/tex]，其中[tex=0.857x1.286]2vV9/H0OKYsecuK6l8Y0Sw==[/tex]是矩阵[tex=0.786x1.0]sHo1pKm+gjxjcUAJjHrarQ==[/tex]的第[tex=0.429x1.214]rmIPPJrP+tFN2kAYPlU/4g==[/tex]列形成的[tex=0.929x0.786]VF0GLe2VBE/4VKNzpyOfFg==[/tex]维列向量，[tex=5.643x1.214]mV8fgjtWrhwph5ozut3E1x6ZrjbWA0WJIpvx9ZvPCkY=[/tex]。另记[tex=8.214x1.357]vmxGLOK1MpFid2XsyYPonGKyP20QYqxCX7XwnFWgD4G4jhdKcsuXTzO3zRccu3GzI0hIqR5/E1mwsKYtqe5vhw==[/tex]，其中[tex=0.929x1.0]XbnFtnwYYYhDa5rLhMoEVQ==[/tex]是矩阵[tex=0.857x1.0]KGogyvwDAIJf/iL0H/9wjg==[/tex]的第[tex=0.429x1.214]rmIPPJrP+tFN2kAYPlU/4g==[/tex]列形成的[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]维未知列向量[tex=5.143x1.214]r7ywkH12Qy/X5A7rZAoVNqiHAvOqHuxEjQ7eVIyM0b0=[/tex]。考虑矩阵方程[tex=14.571x1.357]Xa1ECE2gAMaLlumWDk5aGZVFpqtOzERimL5cRZ5BtWKdm51nVchKNqJPIXy+NTSL6sVLXs3Egskjo6Cgr2TIhaFbt+Cu7xUoCmd9hBBT/Ig2/VOo7HAArw6rnerMOib/I6ovu7J6EBnxBq4QTeG3TA==[/tex]，[tex=5.643x1.214]R62TrLH16dH52btiUsvh5AMhKWrv/8Ea2fVDBHjaUes=[/tex]。现在对[tex=0.429x1.214]rmIPPJrP+tFN2kAYPlU/4g==[/tex]用归纳法证明，矩阵方程[tex=14.571x1.357]fEAkvzQCggbQUue5L0a8Na08kiKQLlrCR8mMk7QVhdoWf8+YGRGoSGgMnACpFCFdJU8R6rVRAc70974jFG0QHFnguA4wZCXLGJXOmDAY4vKEjOVmrJ4VdblWHYPw1dXjgKTK+oxnMnjok+jdL1H4pg==[/tex]有解的充要条件是[tex=13.357x1.357]uqh+oOvD2P9iqZ7dD7XO1Au5GpPM3U/ETOAOUQD4vNHQnkms0gecUMA/0jyy6en1jXYHVWwrwkkA+Jb0z5PM12iALjRcDydizNFaOcY7SGL/OFXAb2n4gj7W2zfpFy9pO5GYVmEKpVAM1B/bKh0qvg==[/tex]。当[tex=1.714x1.214]tZNUKtQigEi1/oxcaXuw7w==[/tex]时结论显然成立，因为矩阵方程即为[tex=3.5x1.214]ReZhFGw9SU7jD+/yO7e2y9tg7vu1vyVz2E5wRMfk/8A=[/tex]，这是通常的线性方程组。设结论对[tex=1.714x1.214]nYb4RGkfX2KojdaSEdwhuw==[/tex]成立，考虑矩阵方程[tex=14.571x1.357]fEAkvzQCggbQUue5L0a8Na08kiKQLlrCR8mMk7QVhdoWf8+YGRGoSGgMnACpFCFdJU8R6rVRAc70974jFG0QHFnguA4wZCXLGJXOmDAY4vKEjOVmrJ4VdblWHYPw1dXjOjMAJcEI/xx1rxrAAwaH7g==[/tex]。它有解的充要条件是矩阵方程[tex=16.357x1.357]fEAkvzQCggbQUue5L0a8Na08kiKQLlrCR8mMk7QVhdoWf8+YGRGoSGgMnACpFCFdHB/YYQNHiiEs0eij5F504R3g0gmnXS19IwNy+2FoU0LZBlsmP80PRYMhC46U5OwWhVGk5uH0Iad3ESdMQMMTR2m1jvtyPz3cCfCzb+ucvfg=[/tex]和[tex=3.357x1.286]2bD5FHBjHrorm3R9ADarkz2fmJsUOrefIb1/dnQrfdM=[/tex]都有解。由归纳假设，使这两个矩阵方程有解的充要条件是[tex=14.214x1.357]uqh+oOvD2P9iqZ7dD7XO1Au5GpPM3U/ETOAOUQD4vNHQnkms0gecUMA/0jyy6en1jXYHVWwrwkkA+Jb0z5PM12iALjRcDydizNFaOcY7SGKJyP6vx39TnKjeJUhWaSWikKwVV63PFFDqYWWWyo3ZQw==[/tex]和[tex=8.643x1.357]uqh+oOvD2P9iqZ7dD7XO1IyK8iz4mWAd81q9bgqbU63Dv5GRyLwzueCUHJGh2uh1qVQnApiYh0ETdqfum8Q7Gy2dSGxBy+A7o++zpF9djt0=[/tex]。这表明[tex=5.643x1.286]6Jd4PlLmEW9UhTZwTsrSjfjvi8pLdqeXrrBgS97XPozBmI19BLQnnmcImJUIeT/N[/tex]均可由矩阵[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]的列向量的极大线性无关向量组线性表出，因此[tex=5.643x1.286]6Jd4PlLmEW9UhTZwTsrSjfjvi8pLdqeXrrBgS97XPoyL8jmWcIT2X3sydFewb6W6[/tex]。即结论对[tex=0.429x1.214]rmIPPJrP+tFN2kAYPlU/4g==[/tex]也成立。注：从证法二可以看出，解矩阵方程[tex=3.143x1.0]TfRL0Lu8PApzv7GgboFLtw==[/tex]可以归结为解[tex=0.571x1.0]FGGpnaR8m8C48rN8O0c7aw==[/tex]个线性方程组[tex=9.357x1.286]ljz4ZV4TeXiYuPQWFGTsjhzObvosOwd1MxqPzcjnMVMR5pulgveoWWPJJG4FWrTg[/tex]，所以说，矩阵方程是线性方程组的推广。因此，矩阵的行的初等变换 在解矩阵方程中是大有作为的。