Mathematics

Problem

Let

A,B

and

C

be matrices of the same order. a. State the commutative and associative properties for matrix addition. b. Using the general

2\times 2

matrices

A=\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix},\;B=\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix},\;C=\begin{pmatrix}c_{11}&c_{12}\\c_{21}&c_{22}\end{pmatrix},

verify that

(A+B)+C=A+(B+C)

by comparing corresponding entries. c. Give one explicit example of two

2\times 2

matrices

P

and

Q

such that

PQ\neq QP

, thereby illustrating that matrix multiplication is not commutative in general.