Determinant Calculator
Determinant Calculator
Result
The determinant of a matrix is a mathematical function that assigns a unique real number value to every input matrix. It is a scalar value calculated for a given square matrix and serves as a scaling factor for matrix transformations. The determinant of a matrix is a crucial tool for solving systems of linear equations, finding the inverse of a square matrix, and various other mathematical operations. It is important to note that the determinant is defined only for square matrices.
Definition of the Determinant of a Matrix:
The determinant of a matrix is defined as the sum of the products of the elements of any row or column and their corresponding co-factors. This definition holds only for square matrices.
Determinant of a Square Matrix:
The determinant can be calculated for square matrices of any order, such as 2x2, 3x3, 4x4, or n×n, where n is the number of rows (equal to the number of columns). The determinant is essentially a function that maps matrices to real numbers. For a set S of square matrices and a set R of real numbers, the determinant is denoted as f: S → R, where f(x) = y, with x ∈ S and y ∈ R, making f(x) the determinant of the input matrix.
Symbol of Determinant:
The determinant of a square matrix A is commonly denoted as det(A) or |A|. Another symbol for the determinant is Δ.
Minor of an Element in a Matrix:
To find the determinant for individual elements of a matrix, minors are used. Minors are determined by eliminating the rows and columns containing the element in question. For instance, the minor of element 5 in the matrix:
| 2 1 2 |
| 4 5 0 |
| 2 0 1 |
The minor of element 5 is the determinant of the submatrix:
| 2 2 |
| 2 1 |
Calculating this determinant, the minor for element 5 is found to be -2.
Cofactors of an Element in a Matrix:
Cofactors are related to minors and can be calculated using the formula Cij = (-1)^(i+j)Mij, where Cij is the cofactor of the element at row i and column j, and Mij is the minor of the same element. .
Adjoint of a Matrix:
The adjoint of a matrix of order n is defined as the transpose of its cofactors. For a matrix A, the adjoint is represented as Adj(A) and is obtained by transposing the matrix of cofactors.
Transpose of a Matrix:
The transpose of a matrix is denoted as AT or A'. It involves swapping the rows and columns of the matrix, changing the order of the matrix.
Read Here : https://www.mathsedu.in/2023/12/eigenvalue-and-eigenvector-calculator.html
Use following link for previous year question papers of BCA
Use following link for previous year question papers of B.Sc (Computer Science)
FYBSc(Cyber and Digital Science) question paper
Notes of Digital Communication and Networking
