下列命题中一定成立的是()(A)若AB=AC,则B=C.(B)若AB=O,则A=O或B=O.(C)若A≠O,则|A|≠0.(D)若|A|≠0,则A≠O.
(A)反例:设[tex=19.714x2.786]K2vMsZ5TBuB8kq2pfBmYYA+J9DpR7QeeM3L2TnvBnaqhzFJea/mf7BfzeiuNS2qpiJ6ZhL76CAP4n1uyr4Q0REz7HQkaas8ORQ5NiCxA++uvuT+LtCCcOWl3I7lDKpr1w1XTmqX3EtVyxn+pRigAQq8KNSaGlpLc+t1O8EESegppB2RFmIBfQTGodxCqnFdk/a0f5DPJbU4dE3wmoB9IY08LNU2G5UxZZiB7/clkSrB0CVvy980dBj9Jok06zpxJ[/tex].[tex=34.929x2.786]OgVntXYK4O34noTl54/1w7ax4142lSpNUGcPhrT4fgoWROiw2q/BjywA/F3C65QKb2jItXsuI5NqGzqIZSWKrtqHjfS6A12C1mokTsajwLStjwV+Ue2iy02SQeJBLfl3JTlnJArLyRKiYE5tA1eRRX5zlHUwB3JiY3s64C0wn039AIK+1Es0uZgchpY/mkkW8OLMwU0ZRAAGLttEKckNjfAAwz/ccyJ17UJOX1aIe9Muu308Sb77YlXaeuZf+ftyPzcRYjYG2UE1hAmxhZYXTZddXfYlesrHQTyukJlVtHIJyozyXY/DoMkiOwRxPq+zCxS81QeaOU0weA9PFfu+nswS+b00I+gLpeHZOvTle8Q8Q7n70AW3kr1Nyrefp/oB+Y+r1oDXNJDScGtZ1h6CEk5Op5UdncFNIIlyXPyajwVF1xqC66Vprd88O2QtIe590pr1QjVmB/+EW/Q6Ta04jOTE98v0LSNo/FQ3y2drJuI=[/tex].AB=AC,但B≠C.(B)反例:设[tex=13.0x2.786]sSXBpxJWudVpH1R35o4LnOPn73PU4ASwxf3y545mYyudAHKCGXdCjyVbAynOXmDTlJo/7kqr9a89iHrh7SghQpN+B2x0Hc8ZNOtv08xzpzN4lDaLZdkTENV1YmtgByIWYH/AuyqfYbMq8oXDiEkaJqSwLflWprWk2UWOJaQuffk=[/tex].[tex=18.857x2.786]L5ruBYJdYxxi5ljvfCpdzOCuoHO+fiH4OQRQaYCUGw5pdi/Te2Kkfjop0BU9nJoRgvN6YooUC0oAVz+teXO7PVkJ5rMyeL1DGt/KrVbQFWEAFDpTaVAGhe4lc+Tp7eWjvnvB+TvVlDh+J8Xsd4EFLCdxVWujAEpY54R8VizIzjLsiznwcNOkD+1mY7b5g+p7bvgY04PS6zqAF5pBa8SdKdEzoCcwGSpApXOsXHbvwx77HZrzckxUoZS9X60zN3PF[/tex],但[tex=6.214x1.214]1gV5Hs1DojaAnQSyGqd9KfIIT0LpcuXZGGyGMpDTYMs=[/tex].(C)反例:设[tex=8.286x2.786]K2vMsZ5TBuB8kq2pfBmYYDMCNfUwaoM5cFsX9Pxef5F27dPzoHcICQwhe3m4CzlVaKkaOzPB/8m0GYuqODL3k6wJNk/ekupSLAQ2ay7K4UY=[/tex],但[tex=5.929x2.786]075gCzZzsMRb6HYXYk9X9wFv9CCpjSg9gpbZvrUExUjuweBLX4oPWWq7DuSbv2BSuWSEH7/k9FfyxspSNwshrw==[/tex].(D)反证,设[tex=2.571x1.214]8sMqA6WLOfcLEKgU7c39Uw==[/tex]则|A|=0,与|A|≠0矛盾,所以A≠O.故本题应选(D).
举一反三
内容
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【单选题】(2.2 矩阵的运算 ) 设 A 为 n 阶矩阵,下列命题成立的是 () (5.0分) A. 若 A ¹ O ,则 | A | ¹ 0 B. 若 A 2 = O ,则 A = O C. 若 A 2 = A ,则 A = O 或 A = I D. 若 | A | ¹ 0,则 A ¹ O
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设a、b、o都是向量,若ab=0,则a=o或b=o
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若AB=O,则一定有A=O或B=O
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设\( A \)为\( n \)阶方阵,则下列命题成立的是( ) A: 若\( {A^2}{\rm{ = }}O \),则 \( A{\rm{ = }}O \) B: 若\( A{A^T}{\rm{ = }}O \),则\( A{\rm{ = }}O \) C: 若\( {A^2} = A \) ,则 \( A{\rm{ = }}O \)或 \( A = E \) D: 若\( A \ne O \) ,则 \( \left| A \right| \ne 0 \)
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对于同阶方阵A、B、C,下列说法正确的是() A: AB=BA B: 若AB=AC,则B=C C: A(BC)=(AB)C D: 若AB=O,则A=O或B=O