Determinant Properties

Row operations, the product rule (AB) = (A) (B), and as a signed volume factor.

The idea

The properties of the determinant — how it responds to row operations, products, and inverses — all follow from one geometric fact: applying $A$ scales area (in the plane) or volume (in space) by the factor $|\det A|$, and the sign of $\det A$ records whether orientation is preserved or reversed. In the plane the picture is the one in the figure: the columns $(a, c)$ and $(b, d)$ of $A$, drawn from the origin, span a parallelogram, and its area is $|\det A|$.

Each row operation now has a predictable effect. Swapping two rows reverses orientation, so it flips the sign of the determinant. Multiplying a row by $c$ stretches one edge of the spanned box by $c$, so it multiplies the determinant by $c$. Adding a multiple of one row to another is a shear, which slides the box without changing its base or its height, so it leaves the determinant unchanged. Shears also explain the triangular case: they clear the off-diagonal entries without changing the determinant, so a triangular matrix has determinant equal to the product of its diagonal entries.

Composing two maps scales volume by both factors in turn, so $\det(AB) = \det(A)\det(B).$ Taking $B = A^{-1}$ gives $\det(A^{-1}) = 1/\det(A)$. Finally, a matrix with $\det A = 0$ crushes space onto something lower-dimensional, and no matrix undoes that, so $A$ is invertible if and only if $\det A \neq 0$.

Ways to work on it

• Walkthrough. How row operations change , plus triangular and geometric meaning.
• Practice. Compute of a random 2× 2 matrix and decide invertibility.
• Hardest. Find determinants of products and inverses without touching the matrices.

Not sure where to start? Take the ten-question placement test.