# 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.

**Explanation:**

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 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) Laplace’s theorem: An n^{th}-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 = 1a_{ij}A_{ij}, i = 1, 2, …, n

Expansion of the determinant along the jth column is expressed in the form: detA = n∑i = 1a_{ij}A_{ij}, j = 1, 2, …, n

9) 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.

10) 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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