Question

4．已知矩陣A=$[\begin{array}{l}{1}&{1}\\{1}&{1}\end{array}]$，求A¹⁰．

Answer 1

分析 根據(jù)矩陣的乘法分別求得A²，A³，A⁴，…，即可求得A¹⁰．

解答 解：A²=$[\begin{array}{l}{1}&{1}\\{1}&{1}\end{array}]$$[\begin{array}{l}{1}&{1}\\{1}&{1}\end{array}]$=$[\begin{array}{l}{2}&{2}\\{2}&{2}\end{array}]$，
A³=$[\begin{array}{l}{2}&{2}\\{2}&{2}\end{array}]$$[\begin{array}{l}{1}&{1}\\{1}&{1}\end{array}]$=$[\begin{array}{l}{4}&{4}\\{4}&{4}\end{array}]$=$[\begin{array}{l}{{2}^{2}}&{{2}^{2}}\\{{2}^{2}}&{{2}^{2}}\end{array}]$，
A⁴=$[\begin{array}{l}{4}&{4}\\{4}&{4}\end{array}]$$[\begin{array}{l}{1}&{1}\\{1}&{1}\end{array}]$=$[\begin{array}{l}{8}&{8}\\{8}&{8}\end{array}]$=$[\begin{array}{l}{{2}^{3}}&{{2}^{3}}\\{{2}^{3}}&{{2}^{3}}\end{array}]$，
…
∴A¹⁰=$[\begin{array}{l}{{2}^{9}}&{{2}^{9}}\\{{2}^{9}}&{{2}^{9}}\end{array}]$．

點(diǎn)評(píng) 本題考查矩陣的乘法的意義，考查矩陣乘法的運(yùn)算法則，考查計(jì)算能力，屬于基礎(chǔ)題．