The meaning of determinant

Geometrically, the absolute value of the determinant of a $2 \times 2$ matrix represents the area scaling factor of the unit square under the transformation defined by the matrix. For a $3 \times 3$ matrix, it represents the volume scaling factor of the unit cube.

Algebraically, a non-zero determinant indicates that the matrix is invertible, meaning there exists an inverse matrix. This also implies that the system of linear equations represented by the matrix has a unique solution. If the determinant is zero, the matrix is singular (not invertible), and the system of equations either has no solution or infinitely many solutions.