determinant
The determinant is a single number squeezed out of a square matrix, and it has a beautifully visual meaning: it is the factor by which the transformation stretches or shrinks area in 2D, or volume in 3D. If a unit square has area 1, then after the map its image has area equal to det(A).
The sign carries extra news. A positive determinant means orientation is preserved; a negative one means the space got flipped, like a mirror image. So the determinant packs two facts into one number: how much things grow, and whether they were turned inside out.
The headline case is det = 0. That means the transformation flattens space into something lower-dimensional, a line or a point, with zero area or volume. Nothing of that thin result can be un-flattened, which is exactly why a matrix with determinant zero has no inverse.
The first map multiplies every area by 6. The second has identical-direction columns, so it squashes the plane onto a line.
For a 2x2 matrix [[a,b],[c,d]], the determinant is a*d - b*c. Quick test: if it is zero, the two columns lie along the same line and the map collapses the plane.