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vktsn0303
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What is the main idea behind the determinant? What was the main purpose for which it was conceived?
BvU said:Extremely useful for solving systems of equations. History here but I suppose you googled that too ?
The main idea behind determinant is to determine whether a square matrix has an inverse or not. It is also used to find the area of a parallelogram or volume of a parallelepiped in linear algebra.
The purpose of determinant is to simplify calculations involving square matrices and to determine certain properties of the matrix, such as invertibility and volume/area.
Determinant is calculated by using a specific formula depending on the size of the matrix. For a 2x2 matrix, it is calculated by multiplying the top left element by the bottom right element and subtracting the product of the top right and bottom left element. For larger matrices, the Laplace expansion method or Gaussian elimination method can be used.
Determinant is significant in linear algebra because it is used to determine key properties of a matrix, such as invertibility, linear independence of vectors, and change of variables in systems of linear equations. It is also used in various applications such as finding the area/volume of shapes and solving systems of linear equations.
Yes, determinant can be negative. The sign of the determinant depends on the orientation of the basis vectors of the matrix. If the orientation is reversed, the determinant will be negative. In terms of volume/area, a negative determinant indicates a reflection or orientation change in the shape being measured.