Determinant of my matrix is zero but the system still has solutions

Question

Three equations, three unknowns, and the determinant of the coefficient matrix comes out 0. My notes say that means no unique solution, but when I row reduce…

sam_the_temp · Accepted Answer

Both are true and there is no contradiction. det = 0 rules out exactly one thing: a single unique solution. It leaves two possibilities open, no solutions at all, or infinitely many. You found the infinitely many case.

Check your row reduced form. If you have a free variable, your solution set is a line or a plane and every point on it will plug back in correctly. Write the answer parametrically, something like (x, y, z) = (1, 0, 2) + t(2, -1, 1), rather than picking one point and calling it the answer.

psu_paula · Answer

The tool that settles it cleanly is rank. Compare the rank of the coefficient matrix A with the rank of the augmented matrix [A|b]. Equal ranks means consistent, and the number of free variables is (number of unknowns − rank). Different ranks means you have a row like and there is no solution at all.

two_hills_up · Answer

Geometrically: three planes. Unique solution means they meet at one point. Determinant zero means at least one plane is redundant or parallel to the others, so you either get a shared line or nothing. Drawing that once made the algebra stop feeling arbitrary for me.

vacuum_analyze · Answer

Also note that Cramer's rule is dead the moment the determinant is zero, since you would be dividing by it. If your course leaned on Cramer's rule this is exactly the case it cannot handle, and row reduction is the fallback for everything.

Determinant of my matrix is zero but the system still has solutions Concept

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