举一反三
- \(A,B\)均为\(n\)阶方阵,且\(A\)与\(B\)合同,\( R\left( A \right) = 4 \),则\( R\left( B \right) = \) ______
- 设方阵\( A \)和\( B \) 等价,\( A \) 的所有\( k \) 阶子式全为零,则( ) A: \( r\left( B \right) < k \) B: \( r\left( B \right) = k \) C: \( r\left( B \right) \ge k \) D: \( r\left( B \right) \le k \)
- 设`\A`是3阶矩阵,将`\A`的第1列与第2列交换得到`\B`,再把`\B`的第2列加到第1列得`\C`,则满足`\AP=C`的可逆矩阵`\P` ( ) A: \[\left[ {\begin{array}{*{20}{c}}1&1&0\\1&{\rm{1}}&{\rm{1}}\\0&0&1\end{array}} \right]\] B: \[\left[ {\begin{array}{*{20}{c}}1&1&0\\1&0&0\\{\rm{1}}&0&1\end{array}} \right]\] C: \[\left[ {\begin{array}{*{20}{c}}1&{\rm{0}}&0\\1&{\rm{1}}&0\\0&0&1\end{array}} \right]\] D: \[\left[ {\begin{array}{*{20}{c}}1&1&0\\1&0&0\\0&0&1\end{array}} \right]\]
- 5.下列函数中,在其定义域上有最大值和最小值的是()。 A: $f(x)=\left\{ \begin{array}{*{35}{l}} \ln \left| x \right|,\ \ \ x\ne 0 \\ 0,\ \ \ \ \ \ \ \ x=0 \\ \end{array} \right.$ B: $f(x)=\ln \left( \left| x \right|+1 \right)\ x\in [-1,1]$ C: $f(x)=\ln \left| x \right|,\ \ \ x\in [-1,1]\backslash \{0\}$ D: $f(x)=\left\{ \begin{array}{*{35}{l}} \ln \left| x \right|,\ \ \ 0\lt |x|\lt 1 \\ 0,\ \ \ \ \ \ \ \ x=0 \\ \end{array} \right.$
- 已知`\n`阶行列式`\|A|=2`,`\m`阶行列式`\|B|=-2`,则`\m+n`阶行列式\[\left| {\begin{array}{*{20}{c}}0&A\\B&0\end{array}} \right|\]的值为 ()
内容
- 0
\[A = \left[ {\begin{array}{*{20}{c}} 2&2&3\\ 2&3&1\\ 3&4&4 \end{array}} \right]\],且`\BA = A + B`,则矩阵`\B=` ( ) </p></p>
- 1
`\A,B`均为3阶矩阵,下列说法正确的是 ( ) A: \[\left| {\begin{array}{*{20}{c}}O&A\\B&O\end{array}} \right| = \left| B \right|\left| A \right|\] B: `\| A | = a^3`则`\| A + A| = 2a^3` C: `\| A | = 0`的充分条件是`\A`的各行元素之和为零 D: `\| B | = 0`的必要条件是`\B`中有两行元素成比例
- 2
设 \( A \)是 \( 3 \times 3 \)矩阵, \( B \)是 \( 4 \times 4 \)矩阵,且\( \left| A \right| = 1,\,\left| B \right| = - 2, \) 则\( \left| {\left| B \right|A} \right| = \) ______
- 3
设 \( A \)是 \( n \)阶方阵,\( R\left( A \right) = r < n \) ,那么( ) A: \( A \)可逆 B: \( A \)中所有\( r \) 阶子式不为零 C: \( \left| A \right| = 0 \) D: \( A \) 中没有不等于零的\( r \)阶子式
- 4
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\( \left| B \right| > 0 \)