Part 21 : Properties of Determinants

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.

Identity matrices of all orders have determinant 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

Sign of determinant changes on performing odd number of row exchanges

Property 3

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

Assuming lambda is any number

Assuming phi is any number

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.

Assuming a, b and c are three different numbers

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.

First and Third rows are identical

First and Second columns are identical

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 3 is applied in first step and Property 1 is applied in second step

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.

Read Part 22 : Eigenvalues and Eigenvectors

You can view the complete series here
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