证明：矩阵的[tex=0.929x1.071]ye7wVa6THCKv06rO3baM9A==[/tex]型初等行变换（即两行变换）可以通过一些[tex=0.929x1.071]502A3kM9YE+9j/2noEqw2g==[/tex]型与[tex=0.929x1.071]vxzTVGpzraNdAfUuV99UxQ==[/tex]型初等行变换实现.
证明：考虑与第[tex=0.357x1.0]+eJLelx8thmbkEj/Y0iCOw==[/tex]行与第[tex=0.429x1.214]adIpAOtu2Zm0WIyZC7drnQ==[/tex]行互换相应的初等矩阵[tex=2.5x1.357]5wYkER1fPMZab/1buiWPvQ==[/tex]，它可以如下得到：[tex=0.5x1.0]ycRjqHa76IDpEZtluYQxdQ==[/tex][tex=3.857x1.857]t+NQEHINMcUvoSLa3Rsg8kwnpolW49MepE+PKdrFgyLMUCuAXHkbOgzjIjWFS/RSHqcijoEczPBnxjxU/eFFhw==[/tex][tex=15.143x8.214]075gCzZzsMRb6HYXYk9X9/YAb1jLOl3/lPMYR0nKE8GdNwtDTWRVRyzoW6Xjw3ubdOVtoc11z810PuuAanYzIL5lWbYPpgkS7RSc6Cc3vmrVm+d9coCQvmm+LLE/T85yFxRqdqaciVbEcsuKMi/L5I7SwHpI6twRQ4Uc92DpuoH1bJjrie6ZFQXGFX/wTv260H8XoWP1WbUSFlXLfE0eADMss6C3JGG+8wnIhtshwIpgYxKIgJMPK11CVWwSeK4+oTSZYETaE79EHZD5z+cYFtHOyaFmcedVcr1tNfQo5uZp40GbH+qAnyhd+K+UO/RcvTgRmi5E+ZzkLdgxxeQM2g==[/tex][tex=3.643x1.857]x6rqsJq7OeZ4aF2uiCbOcmGgdcgGl3yTJ2idSzzbRi2+kCP8b4HfisdCKCVoH734MBdrbjgi5chhvjXtDdzjVQ==[/tex][tex=17.429x8.214]075gCzZzsMRb6HYXYk9X9/YAb1jLOl3/lPMYR0nKE8Ef1+S8H1hB87c3BMs773P48c1+/PisJyEGmBZ9hK7Q1ERuQdeG3kV3n8356g6LnamnkN39Z/HDHKSeeTITg1PcwMJ3qrrpS1Y3+r6NJsa+jQXBzXuOuO7WRwU8yApIshL6Tuf1HWhsf9E4Jzb0hxleYnIEmjJd/3DzQbOnnkowD7v2HFm+pXOvpUbY0+KBJwCYGk4x2hKUSYRWz/9kihdqHwCF3KoNUYK8DSQ9DQBBg2b7R3JPObTZVpryVGIKgw2uk1kI8yH7y36YE6pzCjItcNMCpEN3rm/B09y3Btsgqg5Rp4dg4fmZcpJ5z9KvmpeQiJ7Jpb5TkGiUu4ziH6P/[/tex][tex=3.857x1.857]t+NQEHINMcUvoSLa3Rsg8kwnpolW49MepE+PKdrFgyLMUCuAXHkbOgzjIjWFS/RSHqcijoEczPBnxjxU/eFFhw==[/tex][tex=17.429x8.214]075gCzZzsMRb6HYXYk9X9/YAb1jLOl3/lPMYR0nKE8Ef1+S8H1hB87c3BMs773P48c1+/PisJyEGmBZ9hK7Q1ERuQdeG3kV3n8356g6LnamnkN39Z/HDHKSeeTITg1PcBMQZ8+zrs9Cw8guu5hXPY7wnVbfLUONr9pnYZDTa20vNHVOA6NVhX3d+d4jj9V4+PwT1w2zevyxrbhh+B9G5H8fSdcizSJgYuMa9drbN5mr7DoNpgGAT/ASfMotFuTKGVYY7NDWvvhytJD15dpLEhVX26PlR85qfXnaoFgPLaFwskgcWf+6KdbqyvFdheQvrtkH8My5gH+QpLYs9UBbO4CmlZS06Tg3nAoBG8rMilSILHL1TFg8n66hyZXmtxyXw[/tex][tex=3.643x1.857]W9ritqw/10X/caKVe9mgtmyw01ApOYAze4w4FdyIZnIl6gDtbutd6KmB9BM+lf361KryMcHKRqwME45crfC65g==[/tex][tex=16.643x8.214]075gCzZzsMRb6HYXYk9X9/YAb1jLOl3/lPMYR0nKE8Ef1+S8H1hB87c3BMs773P48c1+/PisJyEGmBZ9hK7Q1ERuQdeG3kV3n8356g6LnamnkN39Z/HDHKSeeTITg1PcBMQZ8+zrs9Cw8guu5hXPY7jGfxtA4hf8Hqo8As20lLXRPIEB/UESrEjZuYvt/O+ZjhVi0Wilpm2biGmZfYZxMt/0wq4bxWMY3QGBceXK1Xek1mT9fg1x4fOm9Gg9uu/ZbrApggXne/+FCvtgWKlBhu5YsfoR+PigB+ORsKRTHVhDlqZ8rP7aXi4ubI1fXeQLaY3ufmB3ILgzeJeNOLZIPHbu95HbYwX/d/3y+AMnLfqSMT84SZIInVzS+38ajIca[/tex]，因此[tex=20.286x1.357]d9eBVAbFouT5S4SiYXacdrxMwFd5erNdSv6x7RndC+8Wy5OElLgCm7a9l4S+T09t9mfcjRlX5avC/Xc2WLUzsA==[/tex].从而[tex=21.357x1.357]m/zQiI3Wkq9CRjTNFyYqZsQmXddcLaJefW1HBx2uGAxuFCc3Liwq6XfUL4Wn+U8UhQMutpPiQaklv0l7RvjIzw==[/tex].这表明对[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]作两行互换可以通过对[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]作一些[tex=0.929x1.071]502A3kM9YE+9j/2noEqw2g==[/tex]型与[tex=0.929x1.071]vxzTVGpzraNdAfUuV99UxQ==[/tex]型初等行变换实现.
举一反三
- 证明：矩阵的[tex=0.929x1.071]GF6lPkbwM1/sKcTXI74B5g==[/tex]型初等行变换(即，两行互换)可以通过一些[tex=0.929x1.071]buZKswv8GSg4ZQySlQudzA==[/tex]型与[tex=0.929x1.071]9/ixLDjuSXaJpaOE7n+RlQ==[/tex]型初等行变换实现.
- 用初等行变换将下列矩阵化为行阶梯型矩阵，并求其秩。[tex=8.429x5.929]dNSDn9BXm1mpm241IkjqDK1acarp7uEXzYmuVtT7iZZFMMQad53UarVdZOO9LaZclqqiQ+Dr+Zs3UUZe2HPgYmFX/TPm1+WolZFZGBdtfN6KHLQE7oP1GphTOG94oqmPX7zTVY8d97MqHXe0iKz5nomJhQqqkQmREGpHhuJgR1g=[/tex]
- 用矩阵的初等行变换将矩阵[tex=11.0x3.643]C3i3DfkQVo2pOvnj2q0JK8uICAslAc7H+uBU3eOkL5pCh0WRibyvhHMij2q4oiCh39plNQtM0tbVoTgWEIcmp+mHwbWEs3+CB80KUDM4ZsK939VlOjBlhOojGwiWdWFZKz7Z0OrHJSXDoUsh0bza4A==[/tex]化为阶梯形矩阵。
- 设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是数域[tex=0.857x1.0]eMszuSG5by5UfRZVROYp5A==[/tex]上的[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级矩阵[tex=3.214x1.357]iagYNk+QmArrUC5iF4MQ2w==[/tex]证明：如果[tex=2.643x1.357]xnNlsIp2wAAq+OkAnU/oIQ==[/tex]，那么：[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]可以表示成[tex=0.929x1.071]502A3kM9YE+9j/2noEqw2g==[/tex]型初等矩阵[tex=3.929x1.357]oHgxo/VJI2syKm7RzHjvnQ==[/tex]的乘积.
- 用初等行变换将下列矩阵化为行最简形矩阵：[tex=7.857x4.786]jcCMHflCR8OS9TosV6N5vE+ILInEdrNZmLPdu5yGc58x2ThFgeZlIoX9A7qb/G3BsPqADtaTucfoTIvayBE9mmX15MEqtmh5xctTqssz6rNbFi75UX3kKIcBpUtwLNzB[/tex].
内容
- 0
利用矩阵的初等行变换将矩阵 [tex=6.5x3.643]NeoTBlf1CmkUoMf07Si5dAD8ZbdksGO0fA9hs2XXNUaQJ6hik+QdSdzXXL7Do2ZtpnsSmK0ZLOe+TqTc8cQQL0vYoBA6XmfXeaLwdFf/3Cv4X/cEuPU+mDvs22bLGJ6N[/tex] 化为行最简矩阵.
- 1
利用矩阵的初等行变换将矩阵 [tex=8.214x3.929]NeoTBlf1CmkUoMf07Si5dEcn2TG6zebvt/5RMBtlBueOLdtecTu8sHIwlVvCx5xG1F3wf+1E2WFPxgaat9EEH3oOxiLDDIWQ3X6tmhQVoC+upxaIXeHCGW7yzZR/s3d3[/tex] 化为行最简矩阵.
- 2
用初等行变换求下列矩阵的的逆矩阵[tex=4.5x2.786]jcCMHflCR8OS9TosV6N5vB0YNYCYNNOTmW+/Xyaeog6rA0KugjnCjJY4NVvcov0ChJerMdjoodQRv4TpwwBfWw==[/tex];
- 3
用初等行变换求矩阵的逆：[tex=5.714x3.5]De166nmeTkb4C/83+ZZH22dbEjNKrnjmBwm3Jdmz0JML6nfyM/ZKF88uPi11IaLL8LZKvSsUZfWX+2Ur3zXuLOHTRCdjXg9pIa5p6FrrHkwt50sAssx/Pxrmf5KzIrOp[/tex]。
- 4
用初等行变换把下列矩阵化为行最简形矩阵．[tex=7.429x3.643]sqT5ECWErx9+sgJ2TkxNkeXrph9cFwwkrb/Ve0fwplRwoOWSteVt0MQptyeDl7tabfUIP1Ppbv+4YDZM3alBASLCdpsfhgsZEOh/LzWOnQU=[/tex]．