Identity Matrix

In a previous blog, I reviewed the basic properties of 2 x 2 matrices. In today's blog, I generalize the definition of the identity matrix and show some of its properties.

Let me start by defining the Identity Matrix I_n.

Definition 1: I_n: Identity Matrix for n x n matrix



The Identity Matrix for n is an n x n matrix consists of values [a_i,j] where i = the row and j = column where:

i = j → a_i,j = 1

i ≠ j → a_i,j=0

Here are a few examples:






If A_n,n is an n x n matrix, we have:

Property 1: A_n,nI_n = A_n,n

Proof:

(1) Let:



(2) Then, A_n,nI_n =






QED

Property 2: I_nA_n,n = A_n,n

Proof:

(1) Let:



(2) Then, I_nA_n,n =






QED

If A_n,m is an n x m matrix, we have:

Property 3: A_n,mI_m=A_n,m

(1) Let:



(2) Then, A_n,mI_m =






QED

References

• "Identity Matrix", Wikipedia.org