Every transformation stretches or squashes area by some factor. That factor is the determinant. Take the unit square (area 1); after the transformation it becomes a parallelogram, and its new area is the determinant. For a 2×2 matrix the formula is

So a determinant of 3 means "areas triple"; a determinant of \tfrac12 means "areas halve". The determinant packs the entire area-effect of a matrix into one number.

Area, live

Move the columns of the matrix (where \hat{\imath} and \hat{\jmath} land). The shaded parallelogram is the image of the unit square, and its area — the absolute determinant — is read off below. Notice the sign: when the columns swap their clockwise order, the determinant goes negative, meaning the plane has been flipped over.

What the sign and size tell you

• Size — how much areas grow or shrink.
• Positive — orientation preserved (no flip).
• Negative — orientation reversed (a reflection is baked in).
• Zero — the square is crushed flat to a line or point; area is destroyed.

That last case is the most important of all: a determinant of zero means the transformation collapses space, and it's the line between matrices you can undo and matrices you can't.