设[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]为[tex=2.714x1.071]Xa6YzCV9VTlW9p4lLOpktw==[/tex]阶方阵，[tex=0.714x1.0]YiLkHgl7MlxE+QjUplQUKA==[/tex]是[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶可逆矩阵, 矩阵[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]的秩为[tex=0.5x0.786]U5O66aolbR1y5vuKrQbXNA==[/tex], 矩阵[tex=3.071x1.0]PxoG+lJftcaSXuD7xhU13Q==[/tex]的秩为[tex=0.857x1.0]5o/cLuWaJfzEVwUboXrosw==[/tex].试证[tex=2.071x1.0]USs9GFT0Wu9uFkvPUS/nkA==[/tex].
举一反三
- 证明若 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 是 [tex=0.929x0.786]VF0GLe2VBE/4VKNzpyOfFg==[/tex] 阶可逆矩阵, [tex=0.857x1.0]PvQ1rNj9zmhWbdNmDhnQhA==[/tex] 是 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶矩阵, [tex=0.786x1.0]sHo1pKm+gjxjcUAJjHrarQ==[/tex] 为 [tex=2.714x1.071]Xa6YzCV9VTlW9p4lLOpktw==[/tex] 矩阵, [tex=0.714x1.0]YiLkHgl7MlxE+QjUplQUKA==[/tex] 为 [tex=2.714x1.071]/nWgWZWXmeNCPcwAggrwNg==[/tex] 矩阵, 则[tex=12.571x2.786]1xLK2S2fjz/DkWdie5OKhUlchjKwuTIGyV4W5lG6BObL/rAGzSN2lXq15WcXL21srEWIPUboONrjoYDzCvlGDdUFsoP4cKGsLaVn/PiaTYXlDYWekXhXYTShEbQntp433iUctOOzyycrYInxUXbE1A==[/tex]若 [tex=0.857x1.0]PvQ1rNj9zmhWbdNmDhnQhA==[/tex] 可逆 (这时 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 不必假定可逆), 则有[tex=12.143x2.786]1xLK2S2fjz/DkWdie5OKhS8T5x62iWwcCK5Ru8KZWv9qLfo3P42WhXEdHMy/L5Cja7m2MuAyN8cg3pMwXJxzCZvkIwXq6vHn/VE2yYikKCAA9Rt7JgSf+0T2BDJkc1HWfssjf5E8MCnmyHdp44UsEQ==[/tex]
- 设矩阵 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 为[tex=2.714x1.071]Xa6YzCV9VTlW9p4lLOpktw==[/tex]矩阵, [tex=0.929x1.0]GTnOCR9hNPsOuxGSyBGTAE4D+bwdNZdKWKqAkIkho7A=[/tex]为[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶矩阵. 已知[tex=3.429x1.357]SMB0AC6IZNDjxg6K+6zWVn+BWIvVutF9O8pSdcF38cg=[/tex], 试证:若 [tex=3.071x1.0]TNRWo7OzENdr5HSXo87j8Q==[/tex], 则 [tex=2.286x1.0]Rm+ZbMwYAAWgp04m4WOymg==[/tex]
- 设矩阵 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 为[tex=2.714x1.071]Xa6YzCV9VTlW9p4lLOpktw==[/tex]矩阵, [tex=0.929x1.0]GTnOCR9hNPsOuxGSyBGTAE4D+bwdNZdKWKqAkIkho7A=[/tex]为[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶矩阵. 已知[tex=3.429x1.357]SMB0AC6IZNDjxg6K+6zWVn+BWIvVutF9O8pSdcF38cg=[/tex], 试证:若 [tex=3.071x1.0]9p6jQHnicI+OkelBMty3Kw==[/tex], 则[tex=2.643x1.0]ePSuKq512kV1g5Dvpe7S7g==[/tex]
- 设[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]是秩为[tex=0.5x0.786]U5O66aolbR1y5vuKrQbXNA==[/tex]的[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶方阵。证明：[tex=2.714x1.214]WdfA4r+NIP6SrigtboMApA==[/tex]的充要条件是存在秩为[tex=0.5x0.786]U5O66aolbR1y5vuKrQbXNA==[/tex]的[tex=2.286x1.071]v8laF85U0CrctV02ZYMlSw==[/tex]矩阵[tex=0.714x1.0]YiLkHgl7MlxE+QjUplQUKA==[/tex]，使得[tex=3.071x1.0]Y7YH5APubJnh2Rl9VeXpqQ==[/tex]，[tex=3.071x1.214]WP/Qz1f8CpnrYoJVgzQjug==[/tex]， 其中[tex=0.786x1.214]DXzCqUwOzWetPe5F0tZBJQ==[/tex]为[tex=0.5x0.786]U5O66aolbR1y5vuKrQbXNA==[/tex]阶单位阵。
- 设 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 为 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶幂零方阵，[tex=0.786x1.0]sHo1pKm+gjxjcUAJjHrarQ==[/tex] 为 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶可逆方阵，且 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 与 [tex=0.786x1.0]sHo1pKm+gjxjcUAJjHrarQ==[/tex] 可换，则 [tex=5.071x1.214]RN2thfSI1MmKxRcibVWDuJHiSryPX2cHjTCV9twFdmY=[/tex] 都是可逆矩阵.