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Problems
Let
A
=
(
k
1
1
1
k
1
1
1
k
)
,
B
=
(
6
6
6
)
.
A=\begin{pmatrix}k & 1 & 1\\1 & k & 1\\1 & 1 & k\end{pmatrix},\qquad B=\begin{pmatrix}6\\6\\6\end{pmatrix}.
A
=
k
1
1
1
k
1
1
1
k
,
B
=
6
6
6
.
…
Mathematics
Problem
Let
A
=
(
k
1
1
1
k
1
1
1
k
)
,
B
=
(
6
6
6
)
.
A=\begin{pmatrix}k & 1 & 1\\1 & k & 1\\1 & 1 & k\end{pmatrix},\qquad B=\begin{pmatrix}6\\6\\6\end{pmatrix}.
A
=
k
1
1
1
k
1
1
1
k
,
B
=
6
6
6
.
(i) Show that
∣
A
∣
=
(
k
−
1
)
2
(
k
+
2
)
|A|=(k-1)^2(k+2)
∣
A
∣
=
(
k
−
1
)
2
(
k
+
2
)
. (ii) Find all real values of
k
k
k
for which
A
A
A
is invertible. (iii) For
k
=
2
k=2
k
=
2
, compute
A
−
1
A^{-1}
A
−
1
. (iv) Using the result of part (iii), solve the system
A
X
=
B
AX=B
A
X
=
B
when
k
=
2
k=2
k
=
2
.
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