Properties of determinants

There are some properties of Determinants, which are commonly used

Property 1

The value of the determinant remains unchanged if it’s rows and

columns are interchanged

(i.e. |𝐴 ^𝑇 | = |A|)

Check Example 6

Property 2

If any two rows (or columns) of a determinant are interchanged then sign of determinant changes

Check Example 7

Property 3

If all elements of a row (or column) are zero, determinant is 0.

Property 4

If any two rows (or columns) of a determinant are identical, the value of determinant is zero.

Check Example 8 for proof

Property 5

If each element of a row (or a column) of a determinant is multiplied by a constant k, then determinant’s value gets multiplied by k

Check Example 9

Property 6

If elements of a row or column of a determinant are expressed as

sum of two (or more) terms,

then the determinant can be expressed as sum of two (or more) determinants.

Check Example 10 for proof

Property 7

If in a determinant all the elements above or below the diagonal is

zero,

then value of the determinant is equal to product of the diagonal elements.