The n × n identity matrix, denoted I or In, is the matrix whose iith entry is 1 and whos...

The n × n identity matrix, denoted I or I_n, is the matrix whose iith entry is 1 and whose other entries are

all zero. That is,

(a) Explicitly write out I₂, I₃, and I₄.

(b) The reason I is called the identity matrix is that it behaves as follows: Let A be any m × n matrix. Then

i. AI_n = A.

ii. I_m A = A.

Prove these results. (Hint: What are the ijth entries of the products in (i) and (ii)?)