# Part 21 : Properties of Determinants

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We have solved determinants using Laplace expansion but by leveraging the properties of determinants, we can solve determinants much faster.

## Property 1

Determinant of any Identity matrix is equal to 1.

## Property 2

If we perform row exchanges (as in Gaussian Elimination), the sign of determinant also changes.

Suppose determinant of matrix A is equal to 2 then

## Property 3

Common multiples of elements in a single row could be taken out of determinant as a constant.

If more than one multiple is taken common then we can multiply them.

## Property 4

If elements in a row or column are expressed as sums then they could also be expressed as sum of determinants.

This property is also applicable for elements expressed as differences.

## Property 5

If more than one rows or columns are identical then determinant is 0.

## Property 6

Row or column of zeros implies that determinant is equal to 0.

## Property 7

For a diagonal matrix the product of elements in diagonal is equal to determinant of matrix. This could be validated using properties 1 and 3.

## Property 8

For a singular matrix the determinant is equal to 0. Singular matrices do not have inverses.

## Property 9

Product of determinants could be expressed as determinant of product of matrices.

## Property 10

Determinant of transpose of any matrix is equal to determinant of original matrix.

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