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Mathematics
Mathematics
Problem
Let
A
A
A
be a
3
×
3
3 \times 3
3
×
3
real matrix such that
A
(
1
0
1
)
=
2
(
1
0
0
)
,
A
(
−
1
0
1
)
=
4
(
−
1
0
1
)
,
A
(
0
1
0
)
=
2
(
0
1
0
)
.
A \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = 2 \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, A \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix} = 4 \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}, A \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = 2 \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}.
A
1
0
1
=
2
1
0
0
,
A
−
1
0
1
=
4
−
1
0
1
,
A
0
1
0
=
2
0
1
0
.
Then, the system
(
A
−
3
I
)
(
x
y
z
)
=
(
1
2
3
)
(A-3I) \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}
(
A
−
3
I
)
x
y
z
=
1
2
3
has (a) exactly two solutions (b) unique solution (c) infinitely many solutions (d) no solution
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