Why shift a matrix before looking for an eigenvector?
A zero determinant signals collapse into a lower dimension. Applying that test to the shifted matrix explains how a nonzero vector can lie in its null space.
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Which equations survive a collapse in dimension? →Once a transformation flattens space, reachable outputs lie in its image. A target inside that image and one outside it lead to different answers.Which outputs disappear when a transformation collapses space? →Some target vectors fall outside the image after a collapse. In the eigenvalue example it is A minus lambda times the identity that becomes singular; this does not make every system unsolvable.How do rotations change in four dimensions? →Compare a fixed axis in three dimensions with planes of rotation in four. The familiar axis description does not carry over unchanged.
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A determinant of 0 would mean that all of space is squished onto something with 0 volumemake this determinant zero, meaning the tweaked transformation squishes space into a lower dimension