Mathematics

Problem

Answer the following questions about zero divisors in matrix multiplication. a. (i) Construct two non-zero matrices

U

and

V

of order

2

with integer entries chosen only from the set

\{1, -1, 0\}

such that

UV

is the zero matrix. b. (ii) Verify by direct calculation that your matrices in part (i) satisfy

UV=O_{2\times2}

. c. (iii) Prove that it is impossible to find two non-zero real numbers

a

and

b

(that is,

1\times1

matrices) whose product is zero.