设 \( A,B \)均为 \( n \)阶方阵,则 \( A = O \)的充要条件是( )
A: \( {A^2} = O \)
B: \( \left| A \right| = 0 \)
C: \( B \ne O \)且\( AB = O \)
D: \( \left| B \right| \ne 0 \)且\( AB = O \)
A: \( {A^2} = O \)
B: \( \left| A \right| = 0 \)
C: \( B \ne O \)且\( AB = O \)
D: \( \left| B \right| \ne 0 \)且\( AB = O \)
举一反三
- 设\( A \) ,\( B \)为\( n \)阶方阵,满足关系 \( AB = O \),则必有( ) A: \( A = B = O \) B: \( A + B = O \) C: \( \left| A \right| = 0 \)或\( \left| B \right| = 0 \) D: \( \left| A \right| + \left| B \right| = 0 \)
- 若\( A,B \)为同阶方阵,且满足\( AB = O \),则有( ) A: \( A = O \)或\( B = O \) B: \( \left| A \right| = 0 \)或\( \left| B \right| = 0 \) C: \( {\left( {A + B} \right)^2} = {A^2} + {B^2} \) D: \( A \)与\( B \)均可逆
- 设\( A,B \)为\( n \)阶方阵,且\( AB{\rm{ = }}O \) ,则必有( ) A: \( BA{\rm{ = }}O \) B: \( A{\rm{ = }}O \)或\( B{\rm{ = }}O \) C: \( A,{\kern 1pt} {\kern 1pt} \,B \)的秩均小于 \( n \) D: \( \left| A \right| = 0 \)或\( \left| B \right| = 0 \)
- 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 \( \left| A \right| \ne 0 \)
- 设\( n \)阶方阵\( A,B \) 满足\( A\left( {B - E} \right) = O \),则必有( ) A: \( A = O \)且\( B = E \) B: \( A + B = E \) C: 当\( \left| A \right| = 0 \) 时, \( B = E \) D: \( \left| A \right| = 0 \)或\( \left| {B - E} \right| = 0 \)