设三阶矩阵[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]满足[tex=6.5x1.286]fKMuxXvMmkhZ6KBFJApU0RzCv0uzUPjL4nc3K7Aig7k=[/tex]，证明：[tex=3.214x1.286]CxRh2MuWfyX9bh9PvBg47Q==[/tex]可逆 .

2022-06-11

设三阶矩阵[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]满足[tex=6.5x1.286]fKMuxXvMmkhZ6KBFJApU0RzCv0uzUPjL4nc3K7Aig7k=[/tex]，证明：[tex=3.214x1.286]CxRh2MuWfyX9bh9PvBg47Q==[/tex]可逆 .

答案：

[b]证明[/b]     已知矩阵方程可变形为[tex=10.5x1.286]rE8xmALQr/snTBrwA/EOCJrN0ezsuaXq2lQME+Ui1YU=[/tex]，所以[tex=3.214x1.286]CxRh2MuWfyX9bh9PvBg47Q==[/tex]可逆 . 

举一反三

内容

0
设[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]均为[tex=0.643x1.286]ZsZs11iKEvfmzDIurZth8g==[/tex]阶实对称矩阵，证明[tex=0.786x1.286]pi/GsQ3apuRt43V3XQq/tA==[/tex]与[tex=0.786x1.286]q1djlrfSWHAqH21hBgtrSw==[/tex]相似的充分必要条件是[tex=0.786x1.286]pi/GsQ3apuRt43V3XQq/tA==[/tex]与[tex=0.786x1.286]q1djlrfSWHAqH21hBgtrSw==[/tex]有相同的特征值 .
1
设[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]为[tex=0.643x1.286]ZsZs11iKEvfmzDIurZth8g==[/tex]阶可逆方阵，则下列等式[input=type:blank,size:6][/input]是错误的 . 未知类型：{'options': ['[tex=7.429x1.429]ZTbiE9Y0ln8So02J2X7smkTz/PgJS9VvpDAl2g7eDKe4gUsAKaYJSvvmmdwi991cMhzFu4GWOcwilAu2g8olqg==[/tex]', '[tex=10.214x1.286]04xA38H7ZVpgg5WrFSMT4c+yjNPApMmkB0rqDvrzHE6AckU+38dJ7SukdONxO1iB0rsStCQG3JPOtjULccUW+btakBj3/UEtJt6Oh2Iz4jU=[/tex]', '[tex=10.571x1.286]28EdYWOVyrJG8r1hi/1Mc5lk2MX3V4h/B2YuW1ABHAap6J2G1MEnV2RRJxoh8pUQ[/tex]', '[tex=6.0x1.286]hm4++QrJQp91sVYV7ruRyhFZzMJEHI5UdUekobEg1A6FpDS5M/s6oj7+Oc9l5cKf[/tex]'], 'type': 102}
2
设[tex=0.643x1.286]ZsZs11iKEvfmzDIurZth8g==[/tex]阶方阵[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]对应的伴随矩阵分别为[tex=2.786x1.286]Ih6DdOteTDQD0IahSuDV+MNSgXPTuu1Ja8QyRKdAvZY=[/tex]，分块矩阵[tex=6.714x2.786]Nw/iCPiBfAazOGg7nvEajlIw5jl+RO2SmNA9Qn1esVrtkE61ENUIdDPBRH6LlIjO5nrmgJINYZl9Mz/vn8A77g==[/tex]，则[tex=0.786x1.286]TKU5UzNEMzEJwORo6mbEYA==[/tex]的伴随矩阵[tex=2.286x1.286]u8qm2vC3bRLP5hGbGpXdxQ==[/tex][input=type:blank,size:6][/input] . 未知类型：{'options': ['[tex=8.571x3.357]075gCzZzsMRb6HYXYk9X9990UXSwhC4aS30j1YPiTgm3hmsejR/bK2E8n1cVIljP7h60RdT5WNUgHpIYQoHRjpKcCKdCYJ5JVFXKNxTEYz8=[/tex]', '[tex=8.571x3.357]075gCzZzsMRb6HYXYk9X987kyvm6Bsv7fv0XQqco2wEnlTNuea0WEb/NW+27Jyq1Wafep6VDoei2pagyRySH9WiMqBoCRNZdNWUvJBmha+4=[/tex]', '[tex=8.571x3.357]075gCzZzsMRb6HYXYk9X91WL96pCMLEUzVoc6/gfhbPbPJbu++pGgO7iJGvPdZgyFUfh2R3zWya9FQcxBpm2BTnPH+YrqimmpYUrtpd7UoA=[/tex]', '[tex=8.571x3.357]075gCzZzsMRb6HYXYk9X9+YJAcVgyDBoIsVPDf3JvYmjIdJkhb05eI5c1c7O7acymrJ1qew3k4pHA1Hx97MhOcjokDjvPmd3LFt0w27+ji8=[/tex]'], 'type': 102}
3
设[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]为同阶对称矩阵，则下列结论中不正确的是[input=type:blank,size:6][/input] . 未知类型：{'options': ['[tex=2.714x1.286]HqIzVk/FgrqCQ/miBSp3sA==[/tex]也是对称矩阵', '[tex=1.571x1.286]cHJ4KDAad01mWuGaiQQpfA==[/tex]也是对称矩阵', '[tex=4.143x1.286]4dNstrEO5SrY9Vx3vH1DNSwdl6C2OGg7J2yuAMxhmBM=[/tex]（[tex=0.857x1.286]VtHyCG+ZQg7fAIyRU+W9ow==[/tex]为正整数）也是对称矩阵', '[tex=5.429x1.286]HADq+tNK5GwsMk1x8KZJs3wpAw6M18PJvuNQqoVPof0=[/tex]也是对称矩阵'], 'type': 102}
4
已知 [tex=2.071x1.286]Lw1m7LuL1SjurX1WRZuPUg==[/tex] 为3阶矩阵，且满足 [tex=7.714x1.286]ThWWPhndKkz3UClWdkawSBreeT7S5i5PvnTbUSWKYOOv9U2rV7yWwLvdAXkYzsfH[/tex]，其中 [tex=0.786x1.286]KdMX/vrMoFuyctoWaUWH8w==[/tex] 是3阶单位矩阵。(I) 证明：矩阵 [tex=3.286x1.286]7/dUziihQFEuopQUmAB3jtRjn7Bmun7c4UQbytj87b8=[/tex] 可逆；(II) 若 [tex=7.571x4.786]174MZEe/izWSafpCRvJbd3cQKHCzrrjGGKpSfjzsHHVXpVP4uwNKwm6JKWYSK3g5xlXwIaRNk+2zOOmSaTeVcClZXyEuVtudF/ZEztSsKpA=[/tex]，求矩阵 [tex=0.857x1.286]BQkHOimMmPUuGqQUunHC8A==[/tex]。（本题满分6分）