Question

1．若矩陣$（\begin{array}{l}{{a}_{11}}&{{a}_{12}}\\{{a}_{21}}&{{a}_{22}}\end{array}）$滿足：a₁₁，a₁₂，a₂₁，a₂₂∈{0，1}，且$|\begin{array}{l}{{a}_{11}}&{{a}_{12}}\\{{a}_{21}}&{{a}_{22}}\end{array}|$=0，則這樣的互不相等的矩陣共有（　�。�

• A． 2個(gè)
• B． 6個(gè)
• C． 8個(gè)
• D． 10個(gè)

Answer 1

分析 根據(jù)題意，分類(lèi)討論，考慮全為0；全為1；三個(gè)0，一個(gè)1；兩個(gè)0，兩個(gè)1，即可得出結(jié)論．

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}，
可得矩陣$（\begin{array}{l}{{a}_{11}}&{{a}_{12}}\\{{a}_{21}}&{{a}_{22}}\end{array}）$可以是$（\begin{array}{l}{0}&{0}\\{0}&{0}\end{array}）$，$（\begin{array}{l}{1}&{1}\\{1}&{1}\end{array}）$，$（\begin{array}{l}{0}&{0}\\{0}&{1}\end{array}）$，$（\begin{array}{l}{1}&{0}\\{0}&{0}\end{array}）$，
$（\begin{array}{l}{0}&{1}\\{0}&{0}\end{array}）$，$（\begin{array}{l}{0}&{0}\\{0}&{1}\end{array}）$，$（\begin{array}{l}{0}&{1}\\{0}&{1}\end{array}）$，$（\begin{array}{l}{1}&{1}\\{0}&{0}\end{array}）$，$（\begin{array}{l}{1}&{0}\\{1}&{0}\end{array}）$，$（\begin{array}{l}{0}&{0}\\{1}&{1}\end{array}）$．
則這樣的互不相等的矩陣共有10個(gè)．
故選：D．

點(diǎn)評(píng) 本題考查二階矩陣，解題的關(guān)鍵是利用二階矩陣的含義，屬于基礎(chǔ)題．