Question

线性代数习题2道，高分求

线性代数习题2道，高分求，要求有过程，不能只给答案

Answer 1

3. α=(1,2,3)^T, β=(1,1/2,1/3)^T, 则 β^T*α=1+1+1=3.
矩阵 A=αβ^T=
[1 1/2 1/3]
[2 1 2/3]
[3 3/2 1]
则 A^2 = αβ^T*αβ^T = α(β^T*α)β^T = 3αβ^T = 3A =
[3 3/2 1]
[6 3 2]
[9 9/2 3]
推广: A^n=3^(n-1)*A.

4. 记 A=(a1, a2, a3)=
[1 1 2]
[2 0 2]
[2 -1 1]
行初等变换为
[1 1 2]
[0 -2 -2]
[0 -3 -3]
行初等变换为
[1 0 1]
[0 1 1]
[0 0 0]
则 a1, a2 为一个最大线性无关组，a3=a1+a2.