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Mathematics
Mathematics
Problem
For a real number
t
t
t
, consider the matrix
T
(
t
)
=
(
1
t
t
2
0
1
t
0
0
1
)
.
T(t)=\begin{pmatrix}1 &t &t^{2}\\ 0 &1 &t\\ 0 &0 &1\end{pmatrix}.
T
(
t
)
=
1
0
0
t
1
0
t
2
t
1
.
(a) Compute
det
(
T
(
t
)
)
\det\bigl(T(t)\bigr)
det
(
T
(
t
)
)
in terms of
t
t
t
. (b) Hence find all real values of
t
t
t
for which
T
(
t
)
T(t)
T
(
t
)
is invertible. (c) For
t
=
1
t=1
t
=
1
, determine
T
(
1
)
−
1
T(1)^{-1}
T
(
1
)
−
1
.
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