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 has both basic and fundamental properties.
Go through the explanation to know about various properties of the example.
Some of the basic properties of determinants are:
1) Suppose In is the Identity matrix of order n n, then det(I) = 1.
2) If the matrix XT is the transpose of matrix X, then det(XT) = 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) Laplace’s theorem: An nth-order determinant can be calculated using the Laplace’s formulas.
Expansion of the determinant along the ith row is given by the formula: det A = n∑j = 1aijAij, i = 1, 2, …, n
Expansion of the determinant along the jth column is expressed in the form: detA = n∑i = 1aijAij, j = 1, 2, …, n
9) Minor: The first minor Mij associated with the element aij 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 jth column.
10) Cofactor: The cofactor Aij is the minor Mij multiplied by (−1) raised to the (i + j) power. Aij = (−1)i + j Mij
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