For the general case, a makes no difference:

Have , which is ,

so, when exists, have and it's solved. But exists iff .

Thus, the general condition that will make that system solvable has nothing to do with the B. It's decided by entirely by M, via .

Using cofactors:

.

So when when

Factoring gives , so when .

Thus this is solveable whenever , no matter what a is.

Now, when , it *might* have solutions, but then the value of will decide that.

Check by doing row reduction on:

When you get:

so (used row 1 to clear column 1)

so

so (used row 2 to clear column 3)

Thus and .

The last equation puts a restriction on a: implies implies .

Thus when s = 1, it has a solution, but only if a = 1/7.

The s = 1 and a = 1/7 solution is: .

I'll leave it to you to see what happens when s = 4.