Let $A = \begin{pmatrix}k & 1\\ 2 & k\end{pmatrix}, \qquad B = \begin{pmatrix}k & -1\\ -2 & k\end{pmatrix},$ …

Mathematics

Problem

Let

A = \begin{pmatrix}k & 1\\ 2 & k\end{pmatrix}, \qquad B = \begin{pmatrix}k & -1\\ -2 & k\end{pmatrix},

where

k

is a real parameter. a. (i) Obtain

AB

and

BA

in terms of

k

. b. (ii) Determine all real values of

k

for which

AB = BA = I_2

. c. (iii) For each admissible value of

k

, verify explicitly that

B

is the inverse of

A

. State the corresponding inverse matrix.