分块矩阵求逆法:下三角形式(C010)

问题

已知,$\boldsymbol{\textcolor{orange}{A}}$, $\boldsymbol{\textcolor{orange}{B}}$ 和 $\boldsymbol{\textcolor{orange}{C}}$ 是元素 的方阵,$\boldsymbol{\textcolor{orange}{O}}$ 是元素 的方阵
则,根据可逆矩阵的性质,$\textcolor{orange}{\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{C} & \boldsymbol{B} \end{array}\right)^{-1}}$ $\textcolor{orange}{=}$ $\textcolor{orange}{?}$

选项

[A].   $\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{C} & \boldsymbol{B} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll} \boldsymbol{A}^{-1} & \boldsymbol{O} \\ \boldsymbol{B}^{-1} \boldsymbol{C} \boldsymbol{A}^{-1} & \boldsymbol{B}^{-1} \end{array}\right)$

[B].   $\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{C} & \boldsymbol{B} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll} \boldsymbol{A}^{-1} & \boldsymbol{O} \\ -\boldsymbol{A}^{-1} \boldsymbol{C} \boldsymbol{B}^{-1} & \boldsymbol{B}^{-1} \end{array}\right)$

[C].   $\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{C} & \boldsymbol{B} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll} \boldsymbol{A}^{-1} & \boldsymbol{O} \\ -\boldsymbol{B}^{-1} \boldsymbol{C} \boldsymbol{A}^{-1} & \boldsymbol{B}^{-1} \end{array}\right)$

[D].   $\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{C} & \boldsymbol{B} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll} \boldsymbol{A}^{-1} & \boldsymbol{O} \\ -\boldsymbol{B} \boldsymbol{C}^{-1} \boldsymbol{A} & \boldsymbol{B}^{-1} \end{array}\right)$


显示答案

$\left(\begin{array}{ll} \boldsymbol{\textcolor{orange}{A}} & \boldsymbol{O} \\ \boldsymbol{\textcolor{yellow}{C}} & \boldsymbol{\textcolor{cyan}{B}} \end{array}\right)^{\textcolor{red}{-1}}$ $=$ $\left(\begin{array}{ll} \boldsymbol{\textcolor{orange}{A}}^{\textcolor{red}{-1}} & \boldsymbol{O} \\ \textcolor{red}{-}\boldsymbol{\textcolor{cyan}{B}}^{\textcolor{red}{-1}} \boldsymbol{\textcolor{yellow}{C}} \boldsymbol{\textcolor{orange}{A}}^{\textcolor{red}{-1}} & \boldsymbol{\textcolor{cyan}{B}}^{\textcolor{red}{-1}} \end{array}\right)$

分块矩阵求逆法:上三角形式(C010)

问题

已知,$\boldsymbol{\textcolor{orange}{A}}$, $\boldsymbol{\textcolor{orange}{B}}$ 和 $\boldsymbol{\textcolor{orange}{C}}$ 是元素 的方阵,$\boldsymbol{\textcolor{orange}{O}}$ 是元素 的方阵
则,根据可逆矩阵的性质,$\textcolor{orange}{\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{C} \\ \boldsymbol{O} & \boldsymbol{B} \end{array}\right)^{-1}}$ $\textcolor{orange}{=}$ $\textcolor{orange}{?}$

选项

[A].   $\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{C} \\ \boldsymbol{O} & \boldsymbol{B} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll} \boldsymbol{A} & -\boldsymbol{A}^{-1} \boldsymbol{C} \boldsymbol{B}^{-1} \\ \boldsymbol{O} & \boldsymbol{B} \end{array}\right)$

[B].   $\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{C} \\ \boldsymbol{O} & \boldsymbol{B} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll} \boldsymbol{A}^{-1} & -\boldsymbol{A}^{-1} \boldsymbol{C} \boldsymbol{B}^{-1} \\ \boldsymbol{O} & \boldsymbol{B}^{-1} \end{array}\right)$

[C].   $\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{C} \\ \boldsymbol{O} & \boldsymbol{B} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll} \boldsymbol{A}^{-1} & -\boldsymbol{A} \boldsymbol{C}^{-1} \boldsymbol{B} \\ \boldsymbol{O} & \boldsymbol{B}^{-1} \end{array}\right)$

[D].   $\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{C} \\ \boldsymbol{O} & \boldsymbol{B} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll} \boldsymbol{A}^{-1} & \boldsymbol{A}^{-1} \boldsymbol{C} \boldsymbol{B}^{-1} \\ \boldsymbol{O} & \boldsymbol{B}^{-1} \end{array}\right)$


显示答案

$\left(\begin{array}{ll} \boldsymbol{\textcolor{orange}{A}} & \boldsymbol{\textcolor{yellow}{C}} \\ \boldsymbol{O} & \boldsymbol{\textcolor{cyan}{B}} \end{array}\right)^{\textcolor{red}{-1}}$ $=$ $\left(\begin{array}{ll} \boldsymbol{\textcolor{orange}{A}}^{-1} & \textcolor{red}{-}\boldsymbol{\textcolor{orange}{A}}^{\textcolor{red}{-1}} \boldsymbol{\textcolor{yellow}{C}} \boldsymbol{\textcolor{cyan}{B}}^{\textcolor{red}{-1}} \\ \boldsymbol{O} & \boldsymbol{\textcolor{cyan}{B}}^{-1} \end{array}\right)$

分块矩阵求逆法:副对角线形式(C010)

问题

已知,$\boldsymbol{\textcolor{orange}{A}}$ 和 $\boldsymbol{\textcolor{orange}{B}}$ 是元素 的方阵,$\boldsymbol{\textcolor{orange}{O}}$ 是元素 的方阵
则,根据可逆矩阵的性质,$\textcolor{orange}{\left(\begin{array}{ll}\boldsymbol{O} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{O} \end{array}\right)^{-1}}$ $\textcolor{orange}{=}$ $\textcolor{orange}{?}$

选项

[A].   $\left(\begin{array}{ll} \boldsymbol{O} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{O} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{cc} \boldsymbol{O} & \boldsymbol{A^{-1}} \\ \boldsymbol{B^{-1}} & \boldsymbol{O} \end{array}\right)$

[B].   $\left(\begin{array}{ll} \boldsymbol{O} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{O} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{cc} \boldsymbol{O} & \boldsymbol{B}^{\top} \\ \boldsymbol{A}^{\top} & \boldsymbol{O} \end{array}\right)$

[C].   $\left(\begin{array}{ll} \boldsymbol{O} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{O} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{cc} \boldsymbol{O} & \boldsymbol{B}^{-1} \\ \boldsymbol{A}^{-1} & \boldsymbol{O} \end{array}\right)$

[D].   $\left(\begin{array}{ll} \boldsymbol{O} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{O} \end{array}\right)^{-1}$ $=$ $\left(\begin{array}{cc} \boldsymbol{B}^{-1} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{A}^{-1} \end{array}\right)$


显示答案

$\left(\begin{array}{ll} \boldsymbol{O} & \boldsymbol{\textcolor{orange}{A}} \\ \boldsymbol{\textcolor{cyan}{B}} & \boldsymbol{O} \end{array}\right)^{\textcolor{red}{-1}}$ $=$ $\left(\begin{array}{cc} \boldsymbol{O} & \boldsymbol{\textcolor{cyan}{B}}^{\textcolor{red}{-1}} \\ \boldsymbol{\textcolor{orange}{A}}^{\textcolor{red}{-1}} & \boldsymbol{O} \end{array}\right)$

分块矩阵求逆法:主对角线形式(C010)

问题

已知,$\boldsymbol{\textcolor{orange}{A}}$ 和 $\boldsymbol{\textcolor{orange}{B}}$ 是元素 的方阵,$\boldsymbol{\textcolor{orange}{O}}$ 是元素 的方阵
则,根据可逆矩阵的性质,$\textcolor{orange}{\left(\begin{array}{ll}\boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{B}\end{array}\right)^{-1}}$ $\textcolor{orange}{=}$ $\textcolor{orange}{?}$

选项

[A].   $\left(\begin{array}{ll}\boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{B}\end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll}\boldsymbol{O} & \boldsymbol{A}^{-1} \\ \boldsymbol{B}^{-1} & \boldsymbol{O}\end{array}\right)$

[B].   $\left(\begin{array}{ll}\boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{B}\end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll}\boldsymbol{B}^{-1} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{A}^{-1}\end{array}\right)$

[C].   $\left(\begin{array}{ll}\boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{B}\end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll}-\boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{O} & -\boldsymbol{B}\end{array}\right)$

[D].   $\left(\begin{array}{ll}\boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{B}\end{array}\right)^{-1}$ $=$ $\left(\begin{array}{ll}\boldsymbol{A}^{-1} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{B}^{-1}\end{array}\right)$


显示答案

$\left(\begin{array}{ll}\boldsymbol{\textcolor{orange}{A}} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{\textcolor{cyan}{B}}\end{array}\right)^{\textcolor{red}{-1}}$ $=$ $\left(\begin{array}{ll}\boldsymbol{\textcolor{orange}{A}}^{\textcolor{red}{-1}} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{\textcolor{cyan}{B}}^{\textcolor{red}{-1}}\end{array}\right)$

用初等变换法求逆矩阵(C010)

问题

已知,$\boldsymbol{E}$ 为单位矩阵,则根据可逆矩阵的性质,以下利用 求逆矩阵的方法表述中,正确的是哪个?

选项

[A].   $\left(\begin{array}{ll}\boldsymbol{A} & \boldsymbol{E}\end{array}\right)$ $\stackrel{\text {初等行变换}}{\longrightarrow}$ $\left(\begin{array}{ll}\boldsymbol{E} & – \boldsymbol{A}^{-1}\end{array}\right)$

[B].   $\left(\begin{array}{ll}\boldsymbol{A} & \boldsymbol{E}\end{array}\right)$ $\stackrel{\text {初等列变换}}{\longrightarrow}$ $\left(\begin{array}{ll}\boldsymbol{E} & \boldsymbol{A}^{-1}\end{array}\right)$

[C].   $\left(\begin{array}{ll}\boldsymbol{A} & \boldsymbol{E}\end{array}\right)$ $\stackrel{\text {初等行变换}}{\longrightarrow}$ $\left(\begin{array}{ll}\boldsymbol{E} & \boldsymbol{A}^{-1}\end{array}\right)$

[D].   $\left(\begin{array}{ll}\boldsymbol{A} & \boldsymbol{E}\end{array}\right)$ $\stackrel{\text {初等行变换}}{\longrightarrow}$ $\left(\begin{array}{ll} \boldsymbol{A}^{-1} & \boldsymbol{E}\end{array}\right)$


显示答案

$\left(\begin{array}{ll}\boldsymbol{\textcolor{orange}{A}} & \boldsymbol{\textcolor{cyan}{E}}\end{array}\right)$ $\stackrel{\text {初等 [行] 变换}}{\longrightarrow}$ $\left(\begin{array}{ll}\boldsymbol{\textcolor{cyan}{E}} & \textcolor{yellow}{\boldsymbol{A}^{-1}}\end{array}\right)$

继续阅读“用初等变换法求逆矩阵(C010)”

用伴随矩阵法求逆矩阵(C010)

问题

已知 $\boldsymbol{A}^{*}$ 为矩阵 $\boldsymbol{A}$ 的伴随矩阵,则根据可逆矩阵的性质,$\textcolor{orange}{\boldsymbol{A}^{-1}}$ $\textcolor{orange}{=}$ $\textcolor{orange}{?}$

选项

[A].   $\boldsymbol{A}^{-1}$ $=$ $|\boldsymbol{A}|$ $\boldsymbol{A}^{*}$

[B].   $\boldsymbol{A}^{-1}$ $=$ $\frac{1}{|\boldsymbol{A}^{*}|}$ $\boldsymbol{A}$

[C].   $\boldsymbol{A}^{-1}$ $=$ $\frac{1}{|\boldsymbol{A}|}$ $\boldsymbol{A}^{*}$

[D].   $\boldsymbol{A}^{-1}$ $=$ $\frac{-1}{|\boldsymbol{A}|}$ $\boldsymbol{A}^{*}$


显示答案

$\textcolor{orange}{\boldsymbol{A}^{-1}}$ $=$ $\frac{\textcolor{cyan}{1}}{\textcolor{green}{|\boldsymbol{A}|}}$ $\textcolor{red}{\boldsymbol{A}^{*}}$

用定义法求逆矩阵(C010)

问题

已知,$\boldsymbol{E}$ 为单位矩阵,则根据可逆矩阵的性质,若 $\textcolor{cyan}{\boldsymbol{A B}}$ $\textcolor{cyan}{=}$ $\textcolor{cyan}{\boldsymbol{E}}$, 则矩阵 $\boldsymbol{A}$ 的逆矩阵 $\textcolor{orange}{\boldsymbol{A}^{-1}}$ $\textcolor{orange}{=}$ $\textcolor{orange}{?}$

选项

[A].   $\boldsymbol{A}^{-1}$ $=$ $- \boldsymbol{B}$

[B].   $\boldsymbol{A}^{-1}$ $=$ $\boldsymbol{B^{-1}}$

[C].   $\boldsymbol{A}^{-1}$ $=$ $\boldsymbol{B}$

[D].   $\boldsymbol{A}^{-1}$ $=$ $\boldsymbol{B B^{-1}}$


显示答案

$\textcolor{orange}{\boldsymbol{A}^{-1}}$ $=$ $\textcolor{red}{\boldsymbol{B}}$

$(\boldsymbol{A}+\boldsymbol{B})^{-1}$ 是否等于 $\boldsymbol{A}^{-1}$ $+$ $\boldsymbol{B}^{-1}$ ?(C010)

问题

根据可逆矩阵的性质,一般情况下,$\textcolor{orange}{(\boldsymbol{A}+\boldsymbol{B})^{-1}}$ $\textcolor{orange}{\boldsymbol{A}^{-1}}$ $\textcolor{orange}{+}$ $\textcolor{orange}{\boldsymbol{B}^{-1}}$ $?$

选项

[A].   $(\boldsymbol{A}+\boldsymbol{B})^{-1}$ $=$ $\boldsymbol{A}^{-1}$ $+$ $\boldsymbol{B}^{-1}$

[B].   $(\boldsymbol{A}+\boldsymbol{B})^{-1}$ $\neq$ $\boldsymbol{A}^{-1}$ $+$ $\boldsymbol{B}^{-1}$


显示答案

$(\boldsymbol{\textcolor{orange}{A}}+\boldsymbol{\textcolor{orange}{B}})^{\textcolor{cyan}{-1}}$ $\textcolor{red}{\neq}$ $\boldsymbol{\textcolor{orange}{A}}^{\textcolor{cyan}{-1}}$ $+$ $\boldsymbol{\textcolor{orange}{B}}^{\textcolor{cyan}{-1}}$

$\left|\boldsymbol{A}^{-1}\right|$ 等于什么?(C010)

问题

根据可逆矩阵的性质,$\textcolor{orange}{\left|\boldsymbol{A}^{-1}\right|}$ $\textcolor{orange}{=}$ $\textcolor{orange}{?}$

选项

[A].   $\left|\boldsymbol{A}^{-1}\right|$ $=$ $|\boldsymbol{A}|$

[B].   $\left|\boldsymbol{A}^{-1}\right|$ $=$ $\frac{-1}{|\boldsymbol{A}|}$

[C].   $\left|\boldsymbol{A}^{-1}\right|$ $=$ $\frac{1}{|\boldsymbol{A}|}$

[D].   $\left|\boldsymbol{A}^{-1}\right|$ $=$ $\frac{1}{|\boldsymbol{A^{-1}}|}$


显示答案

$\left|\boldsymbol{\textcolor{orange}{A}}^{\textcolor{cyan}{-1}}\right|$ $=$ $\frac{\textcolor{red}{1}}{|\boldsymbol{\textcolor{orange}{A}}|}$

$\left(\boldsymbol{A}^{\mathrm{\top}}\right)^{-1}$ 等于什么?(C010)

问题

根据可逆矩阵的性质,$\textcolor{orange}{\left(\boldsymbol{A}^{\mathrm{\top}}\right)^{-1}}$ $\textcolor{orange}{=}$ $\textcolor{orange}{?}$

选项

[A].   $\left(\boldsymbol{A}^{\mathrm{\top}}\right)^{-1}$ $=$ $\left(\boldsymbol{A}^{-1}\right)^{\mathrm{\top}}$

[B].   $\left(\boldsymbol{A}^{\mathrm{\top}}\right)^{-1}$ $=$ $\left(\boldsymbol{A}\right)^{\mathrm{\top}}$

[C].   $\left(\boldsymbol{A}^{\mathrm{\top}}\right)^{-1}$ $=$ $\left(\boldsymbol{A}^{\top}\right)^{\mathrm{\top}}$

[D].   $\left(\boldsymbol{A}^{\mathrm{\top}}\right)^{-1}$ $=$ $\left(\boldsymbol{A}^{-1}\right)^{\mathrm{-1}}$


显示答案

$\left(\boldsymbol{\textcolor{orange}{A}}^{\mathrm{\textcolor{cyan}{\top}}}\right)^{\textcolor{red}{-1}}$ $=$ $\left(\boldsymbol{\textcolor{orange}{A}}^{\textcolor{red}{-1}}\right)^{\mathrm{\textcolor{cyan}{\top}}}$

若 $\boldsymbol{A}$, 则 $\boldsymbol{A}^{\mathrm{T}}$ 是否可逆(C010)

问题

根据可逆矩阵的性质,若矩阵 $\boldsymbol{A}$ 可逆,则矩阵 $\boldsymbol{A}$ 的转置矩阵 $\textcolor{orange}{\boldsymbol{A}^{\mathrm{T}}}$ 可逆?

选项

[A].   

[B].   


显示答案


若 $\boldsymbol{\textcolor{orange}{A}}$ 可逆,则 $\boldsymbol{\textcolor{orange}{A}}^{\mathrm{\textcolor{cyan}{T}}}$

$(\boldsymbol{A B})^{-1}$ 等于什么?(C010)

问题

已知,矩阵 $\boldsymbol{A}$ 和 $\boldsymbol{B}$ 均可逆,矩阵 $\boldsymbol{A B}$ 也可逆,则根据可逆矩阵的性质,$\textcolor{orange}{(\boldsymbol{A B})^{-1}}$ $\textcolor{orange}{=}$ $\textcolor{orange}{?}$

选项

[A].   $(\boldsymbol{A B})^{-1}$ $=$ $(\boldsymbol{B A})^{-1}$

[B].   $(\boldsymbol{A B})^{-1}$ $=$ $-$ $\boldsymbol{B}$ $\boldsymbol{A}$

[C].   $(\boldsymbol{A B})^{-1}$ $=$ $\boldsymbol{A}^{-1}$ $\boldsymbol{B}^{-1}$

[D].   $(\boldsymbol{A B})^{-1}$ $=$ $\boldsymbol{B}^{-1}$ $\boldsymbol{A}^{-1}$


显示答案

$(\boldsymbol{\textcolor{orange}{A} \textcolor{cyan}{B}})^{\textcolor{red}{-1}}$ $=$ $\boldsymbol{\textcolor{cyan}{B}}^{\textcolor{red}{-1}}$ $\boldsymbol{\textcolor{orange}{A}}^{\textcolor{red}{-1}}$

若 $\boldsymbol{A}$, $\boldsymbol{B}$ 可逆,则 $\boldsymbol{A B}$ 是否可逆?(C010)

问题

根据可逆矩阵的性质,若矩阵 $\boldsymbol{A}$ 和 $\boldsymbol{B}$ 均可逆,则矩阵 $\textcolor{orange}{\boldsymbol{A B}}$ 是否 可逆?

选项

[A].   

[B].   


显示答案


若 $\boldsymbol{A}$, $\boldsymbol{B}$ 可逆,则 $\boldsymbol{A B}$ 也可逆

$\boldsymbol{A}^{-1}$ 与 $(k \boldsymbol{A})^{-1}$ 的关系(C010)

问题

已知 $k$ 为常数,且 $k$ $\neq$ $0$, 则根据可逆矩阵的性质,$\textcolor{orange}{\boldsymbol{A}^{-1}}$ 与 $\textcolor{orange}{(k \boldsymbol{A})^{-1}}$ 之间有着怎样的关系?

选项

[A].   $(k \boldsymbol{A})^{-1}$ $=$ $k$ $\boldsymbol{A}^{-1}$

[B].   $(k \boldsymbol{A})^{-1}$ $=$ $\frac{1}{k}$ $\boldsymbol{A}^{-1}$

[C].   $(k \boldsymbol{A})^{-1}$ $=$ $\frac{-1}{k}$ $\boldsymbol{A}^{-1}$

[D].   $(k \boldsymbol{A})^{-1}$ $=$ $\boldsymbol{A}^{-1}$


显示答案

$(\textcolor{orange}{k} \boldsymbol{\textcolor{cyan}{A}})^{\textcolor{red}{-1}}$ $=$ $\textcolor{orange}{\frac{1}{k}}$ $\boldsymbol{\textcolor{cyan}{A}}^{\textcolor{red}{-1}}$

若 $\boldsymbol{A}$ 可逆,则 $k \boldsymbol{A}$ 是否可逆?(C010)

问题

已知 $k$ 为常数且 $k$ $\neq$ $0$, 则根据可逆矩阵的性质,若 $\textcolor{orange}{\boldsymbol{A}}$ 可逆,则 $\textcolor{orange}{k \boldsymbol{A}}$ 是否可逆?

选项

[A].   

[B].   


显示答案


若 $\textcolor{orange}{\boldsymbol{A}}$ 可逆,则 $\textcolor{orange}{k \boldsymbol{A}}$ 也可逆