Why is "the Jacobian" used to refer to matrix and its determinant?

Question

Wolfram states (where $J$ is the Jacobian Matrix):

The determinant of J is the Jacobian determinant (confusingly, often called "the Jacobian" as well) ...

Why would people refer to the Jacobian determinant and the Jacobian Matrix by the same name?

Answer 1

The determinant is often learned about, and used in contexts of calc 3 math when people start learning about multiple integrals and use the Jacobian determinant when performing change of variables.

The actual Jacobian matrix (and to a degree its determinant), has applications outside just performing change of variables. People typically call either "The Jacobian" typically because whether its the Jocobian determinate or Matrix can be deduced from the context on which it is used, so it becomes an "informal" way of addressing it.

Answer 2

From a historical perspective, matrices came relatively late. The Wikipedia article Matrix states:

The term "matrix" (Latin for "womb", derived from mater—mother[106]) was coined by James Joseph Sylvester in 1850

Jacobi died in 1851. Thus, when he invented the Jacobian determinant he did not use the term "matrix". Determinants were known for a long time before that name was used for them. It makes sense to call the particular determinant of Jacobi a "Jacobian" when it is understood that the determinant is being used. Otherwise, the name to use is "Jacobian Matrix".