TRIVIAL AND NONTRIVIAL SOLUTIONS

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TRIVIAL AND NON-TRIVIAL SOLUTIONS Reference
Croft Davision, Chapter 8, Block 4,
p.454-457 http//www.math.utep.edu/sos/math Som
e properties of determinants Consider the system
of equations where a, b, c and d are
constants. Clearly x 0, y 0 is a solution of
the system. We call this
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the trivial solution. But solutions other than
x0 y0 may exist, we call them non-trivial
solutions. Condition for non-trivial
solutions Rewrite the above system of equations
in matrix form
i.e. AX0, where A,X 0 are matrices.
If A 0, the system has non-trivial
solutions. If A ?0, the system has only the
trivial solution.
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Meaning of trivial and non-trivial
solutions Consider For Det(A) 0, means ad -
bc 0 or c ?a, d ?b for some values of
? In this case, the second equation is simply a
multiple of the first. Therefore, depending upon
the value of a, b, c d, the above system has
either i. only the trivial solution ii. an
infinite of non-trivial solutions
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e.g.1 Decide which system has non-trivial
solutions. a. b.
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e.g.2 Decide which system has non-trivial
solutions. a. b.
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e.g.3 Determine the values of ? such that the
following system has non-trivial solutions
Exercise p.457, Q2a, 2b, 2e, 3a, 3b, 3d