设`\A`是`\m \times n`矩阵，`\m` 小于 `\n`，则必有 （ ）
A: \[\left| {{A^T}A} \right| \ne 0\]
B: \[\left| {{A^T}A} \right| = 0\]
C: \[\left| {A{A^T}} \right| > 0\]
D: \[\left| {A{A^T}} \right| < 0\]
A: \[\left| {{A^T}A} \right| \ne 0\]
B: \[\left| {{A^T}A} \right| = 0\]
C: \[\left| {A{A^T}} \right| > 0\]
D: \[\left| {A{A^T}} \right| < 0\]
举一反三
- 设\( A \)为\( n \) 阶方阵, \( B \)是\( A \)经过若干次初等变换后得到的矩阵，则（ ） A: \( \left| A \right| = \left| B \right| \) B: \( \left| A \right| \ne \left| B \right| \) C: 若\( \left| A \right| = 0 \) ，则必有 \( \left| B \right| = 0 \) D: 若\( \left| A \right| > 0 \)，则一定有\( \left| B \right| > 0 \)
- 设 \( A \)为 \( m \times n \)矩阵， \( B \)为 \( n \times m \)矩阵，则下列结论中不正确的是（ ） A: \( {\left( {AB} \right)^T} = {B^T}{A^T} \) B: \( \left| {AB} \right| = \left| {BA} \right| \) C: \( tr\left( {AB} \right) = tr\left( {BA} \right) \) D: \( {A^T}A,\;B{B^T} \)均为\(n\)阶对称阵
- 设\(n\)阶矩阵\(A\)的伴随矩阵为\({A^ * }\),若\(\left| A \right| = 0\)，则\(\left| { { A^ * }} \right| \ne 0\).
- 设向量组\( {\left( {2,1,1,1} \right)^T},{\left( {2,1,a,a} \right)^T},{\left( {3,2,1,a} \right)^T},{\left( {4,3,2,1} \right)^T} \) 线性相关，且\( a \ne 1 \) ，则 \( a = \)______
- 设\( A,\;B \) 均为\( n \) 阶方阵，则必有（ ）. A: \( {(A + B)^2} = {A^2} + 2AB + {B^2} \) B: \( \left| {A + B} \right| = \left| A \right| + \left| B \right| \) C: \( \left| {AB} \right| = \left| A \right|{\kern 1pt} \left| B \right| \) D: \( {\left( {AB} \right)^{\rm T}} = {A^{\rm T}}{B^{\rm T}} \)