In addition to the answers, above, the determinant is a function from the set of set of square matrices into the real numbers that preserves the operation of multiplication: \begin{equation}\det(AB) = \det(A)\det(B) \end{equation} and so it carries $some$ information about square matrices into the much more familiar set of real numbers.

Some examples:

The determinant function maps the identity matrix $I$ to the identity element of the real numbers ($\det(I) = 1$.)

Which real number does not have a multiplicative inverse? The number 0. So which square matrices do not have multiplicative inverses? Those which are mapped to 0 by the determinant function.

What is the determinant of the inverse of a matrix? The inverse of the determinant, of course. (Etc.)

This "operation preserving" property of the determinant explains some of the value of the determinant function and provides a certain level of "intuition" for me in working with matrices.

In addition to the answers, above, the determinant is a function from the set of square matrices into the real numbers that preserves the operation of multiplication: \begin{equation}\det(AB) = \det(A)\det(B) \end{equation} and so it carries $some$ information about square matrices into the much more familiar set of real numbers.

Some examples:

The determinant function maps the identity matrix $I$ to the identity element of the real numbers ($\det(I) = 1$.)

Which real number does not have a multiplicative inverse? The number 0. So which square matrices do not have multiplicative inverses? Those which are mapped to 0 by the determinant function.

What is the determinant of the inverse of a matrix? The inverse of the determinant, of course. (Etc.)

This "operation preserving" property of the determinant explains some of the value of the determinant function and provides a certain level of "intuition" for me in working with matrices.

In addition to the answers, above, the determinant is a function from the set of set of square matrices into the real numbers that $preserves$ the operation of multiplication: \begin{equation}\det(AB) = \det(A)\det(B) \end{equation} and so it carries $some$ information about square matrices into the much more familiar set of real numbers.

Some examples:

The determinant function maps the identity matrix $I$ to the identity element of the real numbers ($\det(I) = 1$.)

Which real number does $not$ have a multiplicative inverse? The number 0. So which square matrices do $not$ have multiplicative inverses? Those which are mapped to 0 by the determinant function.

What is the determinant of the inverse of a matrix? The inverse of the determinant, of course. (Etc.)

This "operation preserving" property of the determinant explains some of the value of the determinant function and provides a certain level of "intuition" for me in working with matrices.

In addition to the answers, above, the determinant is a function from the set of set of square matrices into the real numbers that preserves the operation of multiplication: \begin{equation}\det(AB) = \det(A)\det(B) \end{equation} and so it carries $some$ information about square matrices into the much more familiar set of real numbers.

Some examples:

The determinant function maps the identity matrix $I$ to the identity element of the real numbers ($\det(I) = 1$.)

Which real number does not have a multiplicative inverse? The number 0. So which square matrices do not have multiplicative inverses? Those which are mapped to 0 by the determinant function.

What is the determinant of the inverse of a matrix? The inverse of the determinant, of course. (Etc.)

This "operation preserving" property of the determinant explains some of the value of the determinant function and provides a certain level of "intuition" for me in working with matrices.