這是又一道最近兩天同學們詢問比較多的往年試題。現在給出一種求解方法。
Question1:If
has infinitely solutions, then a is , and b is .
分析：這道題的解法有很多，其中有一種較為本質的解法，就是把增廣矩陣通過初等行變換化為行階梯型，根據行階梯型來給出結果。
Solution:
The system should have freevariable(s) since the system has infinitelysolutions. Thus a – 5 = 0, a = 5 . At the same time, b – 4 = 0, b = 4 .
Furthermore, we can also come to the following conclusions.
i) The linear system has a uniquesolution when a≠ 5 . Since the system has no freevariables when a≠ 5.
ii) The linear system has nosolutions (is inconsistent) when a = 5 and b≠ 4 . By this time, the equation represented by the last row in the row echelon form is 0 x1 + 0 x2 + 0 x3= b – 4 ≠ 0. There are no 3-tuples that could satisfy this equation.
同學們，現在你們會做下面這道題了嗎？
Question2:If a linear system
has no solutions, then a and b .
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