Hello, Given: 2 x 1 + 3 x 2 = x 1 4 x 1 + 3 x 2 = x 1 I need to find all real va

Question

Hello, Given: 2x1 + 3x2 =x1 4x1 + 3x2 =x1 I need to find all real values of, if any, for which the system has a nontrivialsolution. If you are able to help, could you pleaseoffer an explanation or list the steps to solving. Thanks! Hello, Given: 2x1 + 3x2 =x1 4x1 + 3x2 =x1 I need to find all real values of, if any, for which the system has a nontrivialsolution. If you are able to help, could you pleaseoffer an explanation or list the steps to solving. Thanks! I need to find all real values of, if any, for which the system has a nontrivialsolution. If you are able to help, could you pleaseoffer an explanation or list the steps to solving. Thanks!

Explanation / Answer

First, re-write the equations as (2-)x1 + 3x2 = 0 (4-)x1 + 3x2 = 0 It's easy to see that if (2-) is not equal to (4-),then the only solution is the trivial solution. So, for anon-trivial solution to exist, we need to have 2- = 4- 2 = 4 which is impossible So the only solution to the system is the trivial one. ===== You can also do it with matricies. From the re-writtenequations, if the coefficient matrix is invertible, then the onlysolution will be the trivial one. So, we need to find thevalues of that will make the coefficient matrixnon-invertible, which means the values of that make thedeterminant equal to 0. The determinant of the coefficientmatrix is, 3(2-) - 3(4-) = 0 6 - 3 - 12 + 3 = 0 -6 = 0 impossible So, once again, the only solution is the trivial one.

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