Determinant Math Basic Concept

A determinant is a number associated with a matrix. Only SQUARE matrices have a determinant.

The symbol for a determinant can be the phrase “det” in front of a matrix variable, det(A); or vertical bars around a matrix, |A|

Minors and Cofactors:

If A is a square matrix, then the minor of entry aij is denoted by Mij and is defined to be the determinant of the sub-matrix that remains after the i number row and j number column are deleted from A.
The number (−1) i+j Mij is denoted by Cij and is called the cofactor of entry aij.

Determinant (Using cofactor expansion):

If A is an n × n matrix, then the number obtained by multiplying the entries in any row or column of A by the corresponding cofactors and adding the resulting products is called the determinant of A, and the sums themselves are called cofactor expansions of A. That is,

det(A) = a1 jC1 j + a2 jC2 j +… + anj Cnj [Cofactor expansion along the jth column] det(A) = ai1Ci1 + ai 2Ci 2 + … + ainCin [Cofactor expansion along the ith row]

Determinant of a triangular (upper or lower) matrix or diagonal matrix:

If A is an n × n triangular (upper triangular, lower triangular) or diagonal matrix, then det(A) is the product of the entries on the main diagonal of the matrix; that is, det(A) = a11a22 a33…ann .

Properties of Determinants of Matrices:

1. Determinant evaluated across any row or column is same.
2. If all the elements of a row or column are zeros, then the value of the determinant is zero.
3. Determinant of an Identity matrix  (In) is 1.
4. If rows and columns are interchanged then value of determinant remains same (value does not change). Therefore, det(A) = det(A^T), here A^T is transpose of matrix
5. If any two row or two column of a determinant are interchanged the value of the determinant is multiplied by -1.
6. If all elements of a row or column of a determinant are multiplied by some scalar number k, the value of the new determinant is k times of the given determinant. Therefore, if A be an n-rowed square matrix and K be any Then |KA| = Kⁿ|A| .
7. If two rows or columns of a determinant are identical (same) the value of the determinant is zero.
8. Let A and B be two matrix, then det(AB) = det(A)*det(B).
9. If A be a matrix then, |Aⁿ| = |A|ⁿ.
10. Determinant of Inverse of matrix can be defined as |A^-1| = |A|^-1.
11. Determinant of diagonal matrix, triangular matrix (upper triangular or lower triangular matrix) is product of element of the principal.
12. In a determinant each element in any row or column consists of the sum of two terms, then the determinant can be expressed as sum of two determinants of same For example, If B is obtained by adding c-times a row of A to a different row, the det(B) = det(A).
13. If value of determinant Δ becomes zero by substituting x = α, then x-α is a factor of α.
14. Here, cij denotes the cofactor of elements of aij in D.
15. In a determinant the sum of the product of the elements of any row (or column) with the cofactors of the corresponding elements of any other row or column is.. For example, det(A) = ai1Cj1 + ai2Cj2 + ai3Cj3 +…… + ainCjn, here Cj1, Cj2, Cj3 …Cjn are cofactors along elements of jth row.

17. Let λ1 , λ2 , λ3 ,…, λn are the Eigenvalues of A (square matrix of order n). Then det(A) = λ1λ2λ3 …λn , product of Eigenvalues.

DIFFERENCE BETWEEN MATRIX AND DETERMINANT:

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