The determinant of a square matrix provides important information about the associated linear transformation and solutions to systems of linear equations. It can be computed from the entries of the matrix using specific arithmetic expressions. For a 2x2 matrix, the determinant is defined as the product of diagonal entries minus the product of off-diagonal entries. The absolute value of the determinant gives the scale factor by which area or volume is multiplied under the linear transformation, while the sign indicates if orientation is preserved. Determinants occur throughout mathematics and are used in calculus, linear algebra, and as a compact notation for expressions.