What Values of k Affect the Solutions of This Matrix System?

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SUMMARY

The discussion focuses on the values of k that affect the solutions of a specific augmented matrix representing a system of linear equations. The conclusions are definitive: the system has no solutions when k is not equal to 1 or -1, infinitely many solutions when k equals 1, and a unique solution when k equals -1. These findings are crucial for understanding the behavior of linear systems based on parameter variations.

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  • Understanding of augmented matrices in linear algebra
  • Knowledge of the concepts of unique, infinite, and no solutions in linear systems
  • Familiarity with the implications of parameter values on matrix rank
  • Basic skills in solving systems of linear equations
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This discussion is beneficial for students of linear algebra, educators teaching systems of equations, and anyone interested in the mathematical principles governing solution sets in matrix systems.

RajdeepSingh7
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Question :
Consider the following augmented matrix for a system of linear equations.
1 2 3 | 4 
0 1 2  3 |
| 0 0 k − 1  0 |
0 0 0 | k^2−1 

where k is some real number. For which values of k does the system have
(a) no solutions?
(b) infinitely many solutions?
(c) a unique solution?
[ Better version available here ---> http://imageshack.us/photo/my-images/839/screenshot20120416at636.png/ ]


Answer:

a) k is not equal to 1 or -1
b) k = 1
c) k =-1

Just wanted to see if I had it right, and had missed anything, as I had a bit of deliberation for the answers :)
 
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Yes, that is correct.
 

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