Mathematics

Problem

Let

P

and

Q

be square matrices of the same order. a. (i) Define the identity matrix

I_n

. b. (ii) If

PQ = I_n

, does it necessarily follow that

QP = I_n

? Give a brief reason for your answer. c. (iii) Take

P = \begin{pmatrix}2 & -1\\ 1 & 0\end{pmatrix}, \qquad Q = \begin{pmatrix}0 & 1\\ -1 & 2\end{pmatrix}.

Compute

PQ

and

QP

and interpret your results in the light of part (ii).