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Mathematics
Mathematics
Problem
Let
k
k
k
be a real parameter and let
B
(
k
)
=
(
k
1
2
2
k
1
1
2
k
)
.
B(k)=\begin{pmatrix} k &1 &2\\ 2 &k &1\\ 1 &2 &k\end{pmatrix}.
B
(
k
)
=
k
2
1
1
k
2
2
1
k
.
(i) Calculate
det
(
B
(
k
)
)
\det\bigl(B(k)\bigr)
det
(
B
(
k
)
)
in terms of
k
k
k
. (ii) Determine all integer values of
k
k
k
in the interval
[
−
3
,
3
]
[-3,3]
[
−
3
,
3
]
for which
B
(
k
)
B(k)
B
(
k
)
is invertible. (iii) For
k
=
1
k=1
k
=
1
, find
B
(
1
)
−
1
B(1)^{-1}
B
(
1
)
−
1
.
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