tételfüggvény

A tételfüggvény, or determinant function, is a fundamental concept in linear algebra. It is a scalar value that can be computed from the elements of a square matrix. The determinant has numerous applications, including determining the invertibility of a matrix, solving systems of linear equations, and calculating the area or volume of geometric transformations.

For a 2x2 matrix, given by:

[[a, b],

[c, d]]

the determinant is calculated as ad - bc.

For larger square matrices, the determinant can be computed using various methods, such as cofactor expansion or row reduction. Cofactor expansion involves recursively calculating determinants of smaller submatrices. Row reduction involves transforming the matrix into an upper triangular form, where the determinant is the product of the diagonal elements, with adjustments made for row swaps and scalar multiplication.

A matrix has an inverse if and only if its determinant is non-zero. This property is crucial for solving systems of linear equations. If the determinant of the coefficient matrix is zero, the system either has no solution or infinitely many solutions.

The determinant also represents the scaling factor of the linear transformation associated with the matrix. For example, in two dimensions, the absolute value of the determinant of a 2x2 matrix indicates how the area of a region is scaled by the linear transformation. In three dimensions, it represents the scaling factor for volume.