Thread: Finding a value for A so that a system of linear equations is inconsistent

Consider the system of linear equations
x − 3z = −3
−2x − Ay + z = 2
x + 2y + Az = 1


Find the respective values of A such that the system is inconsistent..

I've tried rewriting this as an augemented matrix, then doing row operations to get the bottom row as [0 0 {(A+3)/2 - 5/A} | 2-(4/A)]
so for it to be inconsistent i know that it has to look like [0 0 0 | K] for any real k
so (A+3)/2 - 5/A should equal zero, but 2-4A shouldn't because that would make the system have infinitely many solutions, not NO solutions.

but i can't find any values of A that solve both these equations!

Please help, I'm so confused

Note that the system of linear equations does not have a solution only if the determinant of is 0, the determinant of this matrix is which is 0 at (using quadratic formula) A = -5, A = 2

so to see plug in A = -5, and to verify if the system is inconsistent, first note that By simple examination i see that so the last column vector is linearly dependent, so we can throw it out.

Now is aking to solving

so by observation we note that a solution of this system must have x = -3, which constraints y = so our soln, if we have one is but
which does not equal Thus for A = -5, this system is inconsistent (has no solutions).

Now for A = 2, using the above analysis, it has an infinitely many soln.

So only for A = -5, the system of lin eqn is inconsistent.