What the determinant decides

Determinant and invertibility

The determinant is an invertibility detector. A single number decides the fate of an entire system.

The central theorem

A is invertible   <=>   det(A) ≠ 0

And the exact value of the inverse's determinant follows:

det(A⁻¹) = 1 / det(A)

This result joins the list of equivalences already known:

det(A) ≠ 0
  <=>  A invertible
  <=>  rank(A) = n
  <=>  the columns form a basis
  <=>  A X = 0 has only the zero solution
  <=>  A X = B has a unique solution for every B

Geometrically it is transparent: if det = 0, the transformation flattens space onto an object of lower dimension. And you cannot "un-flatten" — lost information does not come back. No map can reverse a collapse.

The most useful property: the product

det(A × B) = det(A) × det(B)

This is remarkable: the determinant turns a matrix product — a complicated, non-commutative operation — into a plain product of numbers.

The geometric reading makes it obvious: if B multiplies areas by 3 and A by 5, then doing both multiplies them by 15. Order no longer matters here: det(AB) = det(BA), even though AB ≠ BA.

This formula recovers det(A⁻¹) in one line:

A × A⁻¹ = I   ->   det(A) × det(A⁻¹) = det(I) = 1   ->   det(A⁻¹) = 1/det(A)

and confirms in passing that a matrix with zero determinant cannot be invertible: 0 × anything never equals 1.

The other rules

det(tA) = det(A)                 transposing changes nothing
det(λA) = λ^n × det(A)           careful: λ to the power n, not λ
det(I)  = 1

The second one often surprises. Multiplying a 3 × 3 matrix by 2 multiplies its determinant by 2³ = 8: each of the three dimensions is dilated, so volume goes up eightfold.

And what is FALSE

det(A + B) ≠ det(A) + det(B)          in general

There is no simple formula for the determinant of a sum. This is the chapter's most frequent mistake: the determinant is multiplicative, never additive.

Summary