What are determinants in linear algebra?

What are determinants in linear algebra?

Determinants are scalar values that can be computed from the elements of a square matrix. They provide important information about the matrix, such as whether it is invertible.

How do you calculate the determinant of a 2x2 matrix?

For a 2x2 matrix ( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} ), the determinant is calculated as ( \text{det}(A) = ad - bc ).

What does it mean if the determinant is zero?

If the determinant of a matrix is zero, it means the matrix is singular, indicating that it does not have an inverse and its rows or columns are linearly dependent.

Can you explain the geometric interpretation of determinants?

Geometrically, the determinant represents the scaling factor of the transformation described by the matrix. For a 2D matrix, it corresponds to the area of the parallelogram formed by its column vectors.