Mathematics

Problem

Consider the matrices

A = \begin{pmatrix}1 & 2 & 1\\ 0 & 1 & 4\\ 0 & 0 & 1\end{pmatrix}, \qquad B = \begin{pmatrix}1 & -2 & x\\ 0 & 1 & -4\\ 0 & 0 & 1\end{pmatrix},

where

x

is a real number. a. (i) Find the value of

x

for which

AB = I_3

. b. (ii) Using the value of

x

obtained, verify that

BA = I_3

. c. (iii) Comment on the uniqueness of the inverse of a non-singular matrix in the light of your calculations.