The scalar value that is calculated using the elements of the square matrix and which conceals certain properties of a linear transformation of the matrix described is called the determinant of a square matrix. The determinant of a matrix is denoted as det (A) or |A| or det A. The value of determinants may be positive or negative in accordance with the linear transformation being preserved or reversed orientation of a vector space that is real.
Consider a matrix A with four elements. A =
The determinant of matrix A is
Properties of determinants are described below:
1] When the columns and rows of the matrix are interchanged, the sign also changes.
2] If all the multiples of rows and columns are added at once, the determinant value doesn’t change.
3] The scalar value can be obtained from the rows and columns.
4] Say a number other than vector ‘p’ is multiplied to any row of the given matrix, then the value of the determinant is also multiplied by the same value ‘p’.
5] If any one of the columns or rows of a matrix is 0, then its determinant is 0.
6] If any two of the columns or rows of a matrix are 0, then its determinant is 0.
7] If the value of the elements of determinant above and below the principal diagonal are 0, then the value of the determinant is the same as the product of the diagonal elements.
How to reduce determinants?
In order to compute the determinant of a three x three [3 x 3] square matrix or of higher-order involves more time and the process is tedious. The below article answers the question of how do you simplify determinants and reduces the time taken to find the determinant of greater order matrices. The various methods of finding the determinants include the operations on row or column to amend a few entries of the matrix to 0s.
The steps are as follows:
1] From a row or a column, a constant can be factored.
2] The row and column multiples can be added to each other.
3] Any two columns or rows can be interchanged.
4] The determinant can be reduced to its simplest form using its properties.
5] In the case of a lower or an upper triangular matrix, the product of diagonal elements is the same as its determinant value.
6] In order to reduce the value of a determinant to its triangular structure, the method of Gauss elimination can be used.
7] If the row operations are done on a given matrix, it doesn’t alter the determinant value.
9] If the row operations are done on a given matrix, it alters the sign of the determinant.
Few applications of determinants are listed below.
1] The concept of the determinant can be used in determining whether inverse exists for the given matrix and also while calculating the inverse of a matrix using the cofactor method.
2] The method of Cramer’s rule can be used to find the algebraic solutions for the given linear system of equations.
3] The value of the determinant of the vectors that are real is the same as the parallelepiped’s volume which is spanned by the vectors.
4] The determinant of the given matrix is 0 when either the rows or columns of the matrix are dependent linearly.
The concept of determinants occurs throughout mathematics. The various applications of determinants exist, a few are listed above for reference.