Determinants and inverses

Once you see a matrix as a motion of space, two natural questions arise: how much does it stretch or squash space, and can the motion be reversed? The determinant answers the first, and it decides the second.

The determinant as area

Take the unit square (the little square built on the axis arrows) and apply a linear transformation. It becomes a parallelogram. The determinant of the matrix is the factor by which its area changed. A determinant of $2$ means areas double; a determinant of $\frac{1}{2}$ means they halve. For a $2 \times 2$ matrix

\det (\begin{matrix} a & b \\ c & d \end{matrix}) = a d - b c .

In three dimensions it measures how volumes scale, and the idea carries on into higher dimensions.

Sign and collapse

The determinant can be negative, which means the transformation also flips space over, turning it inside out like a reflection. And it can be zero: the crucial case. A zero determinant means the transformation crushes space down onto a line or a point, collapsing a whole dimension. Area becomes nothing.

Invertibility

That collapse is exactly what makes a transformation impossible to undo. If a matrix squashes the plane onto a line, many different starting points land on the same place, and there is no way to tell them apart afterwards: information is lost. So:

$\det \neq 0$ : the transformation is invertible. It can be reversed by another matrix, the inverse $A^{- 1}$ , which undoes it exactly. Every output came from one unique input.
$\det = 0$ : the transformation is not invertible. It has flattened space, and cannot be undone.

Why it matters

This ties directly back to solving $A x = b$ . When $\det A \neq 0$ , the system has exactly one solution, found by applying the inverse: $x = A^{- 1} b$ . When $\det A = 0$ , the system either has no solution or infinitely many, because the transformation has lost the dimension needed to pin the answer down. A single number (the determinant) tells you which world you are in.