標題:
Linear algebra part 2~urgent!!
發問:
Which of the following statements are true and which false? (If false, then give a counter example. If true, then give a brief explanation of at most one sentence)iv. If A is an invertible n x n matrix, then A^m =/ O_(n,n) for all natural m.v. If A is an n x n matrix with A^k = I_n for some k>0, then A... 顯示更多 Which of the following statements are true and which false? (If false, then give a counter example. If true, then give a brief explanation of at most one sentence) iv. If A is an invertible n x n matrix, then A^m =/ O_(n,n) for all natural m. v. If A is an n x n matrix with A^k = I_n for some k>0, then A has an inverse A^-1 = A^(k-1). vi. If A is an n x n matrix with A^6 = I_n, then either A^3 = I_n or A^3 = -I_n.
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最佳解答:
iv True. Otherwise let m be the smallest number such that A^m=0 Then m>=2 Also (A^m-1)A=O A^(m-1)AA^-1=o A^(m-1)=0 Contradiction v True A^k = I AA^(k-1)=I A^(k-1)=A^-1 Since det A not equal 0, the inverse of A exist vi False A^6 = I (A^3)(A^3)=I That means the inverse of A^3 is A^3 itself But this cannot ensure that A^3 = I_n or A^3 = -I_n
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