Let A = pmatrix k & 1 2 & k pmatrix , B = pmatrix k & -1 -2 & k pmatrix

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

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. (i) Obtain

AB

and

BA

in terms of

k

. (ii) Determine all real values of

k

for which

AB = BA = I_2

. (iii) For each admissible value of

k

, verify explicitly that

B

is the inverse of

A

. State the corresponding inverse matrix.