SOLUTION: For the given matrix, the values of h and k are real constants. For which value of h and k are there infinite solutions? no solutions? one solution? [2 1 | h] [4 k | 0]

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Question 1199573: For the given matrix, the values of h and k are real constants. For which value of h and k are there infinite solutions? no solutions? one solution?
[2 1 | h]
[4 k | 0]
Answer by ikleyn(52788)   (Show Source): You can put this solution on YOUR website!
.
For the given matrix, the values of h and k are real constants.
For which value of h and k are there (a) infinite solutions? (b) no solutions? (c) one solution?
[2 1 | h]
[4 k | 0]
~~~~~~~~~~~~~~~~~~~~ 

The corresponding matrix equation system is

    2x + 1y = h

    4x + ky = 0.


The determinant of the coefficient matrix is  d = 2k -4*1 = 2k - 4.

The determinant is zero if and only if d = 0 = 2k-4,  or  k = 2.



Case (a):  The system has infinite solutions if  k = 2  (meaning the determinant is zero)  and  h = 0.


Case (b):  The system has no solutions if  k = 2  (meaning the determinant is zero)  and  h =/= 0.


Case (c):  The system has a unique solution if  k =/= 2  (meaning the determinant is not zero).  
                                                         In this case, there is no restrictions for h.

Solved and answered.



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