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why does det(kA)=(k^n)detA and not kdetA?
This is a fundamental property of determinants, known as the scaling property. It means that when a matrix is multiplied by a constant, the determinant is also multiplied by that constant raised to the power of the matrix's order (or size). In other words, if A is an n x n matrix and k is a constant, then det(kA) = k^n det(A).
One way to prove this property is by showing that it holds for elementary matrices, which are matrices that represent elementary row operations. These operations include swapping two rows, multiplying a row by a constant, and adding a multiple of one row to another row. It can be shown that each of these operations affects the determinant in a predictable way, ultimately leading to the scaling property.
Yes, this property holds for any square matrix, regardless of its size or type. It is a fundamental property of determinants and holds true for all types of matrices, including real, complex, and even infinite-dimensional matrices.
The scaling property of determinants is used in many areas of mathematics, physics, and engineering. For example, it is used to solve systems of linear equations, calculate the inverse of a matrix, and determine whether a linear transformation is invertible. It also has applications in areas such as computer graphics, cryptography, and statistics.
Yes, there are several other properties of determinants that are related to the scaling property. These include the fact that the determinant of the product of two matrices is equal to the product of their determinants, and that the determinant of the transpose of a matrix is equal to the determinant of the original matrix. These properties, along with the scaling property, are essential for understanding and working with determinants in various applications.