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Solution 1
1. Find the determinant det(M) of the 3x3-coefficient matrix M.
The determinant depends on parameter A.
More exactly, the determinant is a linear function of the parameter A.
The necessary condition for the system to be inconsistent is det(M) = 0.
2. Solve this equation det(M) = 0 for A and find its roots.
3. The condition det(M) = 0 is the necessary, but still not the sufficient condition for the system to be inconsistent.
4. So, you need to check for the found value of A whether the system has a solution or no.
But with the found value of A, the system is simply with numerical coefficients.
It just does not contain the parameter.
It contain only numerical coefficients.
This is the plan.
Unfortunately, I don't know your grade and your level of knowledge in the subject.
But my question is: Do you understand this plan?
Are you ready to follow it ?
Do you understand what to do?
There is another way, more easy, in my view.
Solution 2
1. By adding and subtracting equations, eliminate "x" from the system and reduce the given 3x3-system in 3 unknowns
to 2x2-system in 2 unknowns "y" and "z".
The matrix still will contain the parameter A, but it will be only 2x2-matrix.
2. Starting from this point, you need to make the same steps as in the Solution 1.
But this time for 2x2-matrix instead of the original 3x3-matrix, which is MUCH easier.
Please feel free to write me your feedback.
If you need more help, let me know via the "Thank you" message.
But I need to know what is your grade and what is your level of knowledge in the subject.
Do you know the Elimination method ?
Do you know the Substitution method ?
Do you know the Determinant method ?
Do you know what the determinant is and how to calculate it ?
Do you know criterions for a matrix equation to be consistent/inconsistent/dependent ?
Do you know the Cramer's rule ?
It doesn't seem to be a school problem, isn't it?
Am I right ?
Is this the college problem ?
I need to know with whom and for whom I work . . .
Actually, this site www.algebra.com contains lessons that cover all the positions listed above,
and if I see the wishing and the desire from your side to learn it, I can give you the list of lessons.
Good luck !!
