112年 - 112 國立中山大學_碩士班暨碩士在職專班招生考試_應數系碩士班/甲組:基礎數學#130245

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6. Find the point closest to the origin on the curve of intersection of the plane y + 2z = 3 and the cone z² = x² + y².

7. Let T be the endomorphism of R³ defined by . Find the eigenvalues of T and, for each eigenvalue, find the associated eigenvector such that its length equals to 1.

8. Let A = , B = be two matrices with ＞ 0, 1 ≤ i, j ≤ n. Define

Prove that κ(A, B) ≥ 0.

9. Let ≥ 0 for all n ∈ N. Prove that converges if and only if converges.