write the further use of determinant
Answers
- 1)To find the volume of parallelepiped and tetrahedron by vector method
- 2)To state the condition for the equation ax2+2hxy+by2 +2gx+2fy+c =0 representing a pair of straight lines.
- 3)To find the shortest distance between two skew lines.
- 4)Test for intersection of two line in three dimensional geometry.
- 5) To find cross product of two vectors and scalar triple product of vectors by
- 6)Formation of differential equation eliminating arbitrary constant.
Pre-requisite :-
A determinant can be defined in many ways for a square matrix.
The first and most simple way is to formulate the determinant by taking into account the top row elements and the corresponding minors. Take the first element of the top row and multiply it by it’s minor, then subtract the product of the second element and its minor. Continue to alternately add and subtract the product of each element of the top row with its respective minor until all the elements of the top row have been considered.
Properties of Determinant :-
- If In is the identity matrix of the order nxn, then det(I) = 1
- If the matrix M^T is the transpose of matrix M, then det (M^T) = det (M)
- If matrix M^-1 is the inverse of matrix M, then det (M^-1) = 1/det(M) = det (M)^-1
- If two square matrices M and N have the same size, then det (MN) = det (M) det (N)
- If matrix M has a size axa and C is a constant, then det (CM) = C^a det (M)
- If X, Y, and Z are three positive semidefinite matrices of equal size, then the following holds true along with the corollary det (X+Y) ≥ det(X) + det (Y) for X,Y, Z ≥ 0 det (X+Y+Z) + det C ≥ det (X+Y) + det (Y+Z)
- In a triangular matrix, the determinant is equal to the product of the diagonal elements.
- The determinant of a matrix is zero if all the elements of the matrix are zero.
- Laplace’s Formula and the Adjugate Matrix.
USES :-
There are 10 main properties of determinants which include reflection property, all-zero property, proportionality or repetition property, switching property, scalar multiple property, sum property, invariance property, factor property, triangle property, and co-factor matrix property.
1. Reflection Property:
The determinant remains unaltered if its rows are changed into columns and the columns into rows. This is known as the property of reflection.
2. All-zero Property:
If all the elements of a row (or column) are zero, then the determinant is zero.
3. Proportionality (Repetition) Property:
If the all elements of a row (or column) are proportional (identical) to the elements of some other row (or column), then the determinant is zero.
4. Switching Property:
The interchange of any two rows (or columns) of the determinant changes its sign.
5. Scalar Multiple Property:
If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.
6. Sum Property:
7. Property of Invariance:
That is, a determinant remains unaltered under an operation of the form Ci→Ci + αCj + βCk, where j,k ≠ i, or an operation of the form
Ri→Ri + αRj + βRk, where j,k≠i.
8. Factor Property:
If a determinant Δ becomes zero when we put x=α, then (x−α) is a factor of Δ.
9. Triangle Property:
If all the elements of a determinant above or below the main diagonal consist of zeros, then the determinant is equal to the product of diagonal elements. That is,