Question

17．已知矩陣M=$[\begin{array}{l}{2}&{m}\\{n}&{1}\end{array}]$的兩個(gè)特征向量a₁=$[\begin{array}{l}{1}\\{0}\end{array}]$，a₂=$[\begin{array}{l}{0}\\{1}\end{array}]$，若β=$[\begin{array}{l}{1}\\{2}\end{array}]$，求M²β．

Answer 1

分析 由矩陣特征值和特征向量的性質(zhì)，列方程組求得m，n和λ₁，λ₂的值，求得矩陣M，由β=$[\begin{array}{l}{1}\\{2}\end{array}]$=$[\begin{array}{l}{1}\\{0}\end{array}]$+2$[\begin{array}{l}{0}\\{1}\end{array}]$=α₁+2α₂，根據(jù)矩陣乘法及特征值的定義，即可求得M²β的值．

解答 解：設(shè)矩陣M特征向量α₁對(duì)應(yīng)的特征值為λ₁，特征向量α₂對(duì)應(yīng)的特征值為λ₂，
則由$[\begin{array}{l}{2}&{m}\\{n}&{1}\end{array}]$$[\begin{array}{l}{1}\\{0}\end{array}]$=λ₁$[\begin{array}{l}{1}\\{0}\end{array}]$，即$[\begin{array}{l}{2}\\{n}\end{array}]$=$[\begin{array}{l}{{λ}_{1}}\\{0}\end{array}]$，
$[\begin{array}{l}{2}&{m}\\{n}&{1}\end{array}]$$[\begin{array}{l}{0}\\{1}\end{array}]$=λ₂$[\begin{array}{l}{0}\\{1}\end{array}]$，即$[\begin{array}{l}{m}\\{1}\end{array}]$=$[\begin{array}{l}{0}\\{{λ}_{2}}\end{array}]$
∴$\left\{\begin{array}{l}{m=0}\\{n=0}\\{{λ}_{1}=2}\\{{λ}_{2}=1}\end{array}\right.$，
∴矩陣M=$[\begin{array}{l}{2}&{0}\\{0}&{1}\end{array}]$，…（4分）
又β=$[\begin{array}{l}{1}\\{2}\end{array}]$=$[\begin{array}{l}{1}\\{0}\end{array}]$+2$[\begin{array}{l}{0}\\{1}\end{array}]$=α₁+2α₂，…（6分）
∴M²β=M²（α₁+2α₂）=${λ}_{1}^{2}$α₁+2${λ}_{2}^{2}$α₂=4$[\begin{array}{l}{1}\\{0}\end{array}]$+2$[\begin{array}{l}{0}\\{1}\end{array}]$=$[\begin{array}{l}{4}\\{2}\end{array}]$．．…（10分）

點(diǎn)評(píng) 本題考查矩陣的特征值及特征向量的性質(zhì)，考查矩陣乘法的意義，考查計(jì)算能力，屬于中檔題．