Question

1．若矩阵$（\begin{array}{l}{{a}_{11}}&{{a}_{12}}\\{{a}_{21}}&{{a}_{22}}\end{array}）$满足：a₁₁，a₁₂，a₂₁，a₂₂∈{0，1}，且$|\begin{array}{l}{{a}_{11}}&{{a}_{12}}\\{{a}_{21}}&{{a}_{22}}\end{array}|$=0，则这样的互不相等的矩阵共有（　　）

• A． 2个
• B． 6个
• C． 8个
• D． 10个

Answer 1

分析 根据题意，分类讨论，考虑全为0；全为1；三个0，一个1；两个0，两个1，即可得出结论．

解答 解：由 $|\begin{array}{l}{{a}_{11}}&{{a}_{12}}\\{{a}_{21}}&{{a}_{22}}\end{array}|$=0，
可得a₁₁a₂₂-a₁₂a₂₁=0，
由于a₁₁，a₁₂，a₂₁，a₂₂∈{0，1}，
可得矩阵$（\begin{array}{l}{{a}_{11}}&{{a}_{12}}\\{{a}_{21}}&{{a}_{22}}\end{array}）$可以是$（\begin{array}{l}{0}&{0}\\{0}&{0}\end{array}）$，$（\begin{array}{l}{1}&{1}\\{1}&{1}\end{array}）$，$（\begin{array}{l}{0}&{0}\\{0}&{1}\end{array}）$，$（\begin{array}{l}{1}&{0}\\{0}&{0}\end{array}）$，
$（\begin{array}{l}{0}&{1}\\{0}&{0}\end{array}）$，$（\begin{array}{l}{0}&{0}\\{0}&{1}\end{array}）$，$（\begin{array}{l}{0}&{1}\\{0}&{1}\end{array}）$，$（\begin{array}{l}{1}&{1}\\{0}&{0}\end{array}）$，$（\begin{array}{l}{1}&{0}\\{1}&{0}\end{array}）$，$（\begin{array}{l}{0}&{0}\\{1}&{1}\end{array}）$．
则这样的互不相等的矩阵共有10个．
故选：D．

点评 本题考查二阶矩阵，解题的关键是利用二阶矩阵的含义，属于基础题．