What are the Properties of determinants?

In linear algebra, a determinant is a unique number that can be ascertained as a square matrix.

Answer: Determinants have both basic and fundamental properties.

Go through the explanation to know about various properties of the example.

Explanation:

Some of the basic properties of determinants are:

1) Suppose I is the identity matrix of order n, then det(I) = 1.

2) If the matrix X^T is the transpose of matrix X, then det(X^T) = det(X)

3) If X^-1 is the inverse matrix of X then det(X^-1) = 1 / det(X)

4) If two matrices X and Y are of equal size then det(XY) = det(X) det(Y)

5) The determinant of a matrix is zero if each element of the matrix is zero.

6) In a triangular matrix, the determinant is equal to the product of the diagonal elements.

7) Multiplication of a row (column) of a determinant by a constant: Multiplication of the elements of any row (or column) by the same number is equivalent to multiplying the determinant by that number. 

8) Minor: The first minor M_ij associated with the element a_ij of an nth order square matrix A is the determinant of order (n−1) obtained from the matrix A by deleting the ith row and the j^th column.

9) Cofactor: The cofactor Aij is the minor Mij multiplied by (−1) raised to the (i + j) power. A_ij = (−1)^{i + j} M_ij

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