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Step-by-Step Solution
Step 1: Understand the Given Condition
We have a matrix
A = \begin{pmatrix}
a & b \\
0 & d
\end{pmatrix}
with
a, b, d \in \{-1,\,0,\,1\}
and an identity matrix
I = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}.
The condition given is
(I - A)^3 = I - A^3.
Step 2: Expand (I - A)^3
Recall the binomial expansion for
(I - A)^3
is
I - 3A + 3A^2 - A^3.
Therefore:
(I - A)^3 = I - 3A + 3A^2 - A^3.
Step 3: Use the Given Equation
The condition
(I - A)^3 = I - A^3
becomes
I - 3A + 3A^2 - A^3 = I - A^3.
Simplify both sides to get:
- 3A + 3A^2 = 0.
Factor out
3A
to obtain:
3A \bigl(A - I\bigr) = 0 \quad\Longrightarrow\quad A^2 = A.
This tells us that
A
must satisfy
A^2 = A
(i.e.,
A
is idempotent).
Step 4: Compute A^2
Since
A = \begin{pmatrix}
a & b \\
0 & d
\end{pmatrix},
we calculate
A^2 =
\begin{pmatrix}
a & b \\
0 & d
\end{pmatrix}
\begin{pmatrix}
a & b \\
0 & d
\end{pmatrix}
=
\begin{pmatrix}
a^2 & ab + bd \\
0 & d^2
\end{pmatrix}.
For
A^2 = A,
we need
\begin{pmatrix}
a^2 & ab + bd \\
0 & d^2
\end{pmatrix}
=
\begin{pmatrix}
a & b \\
0 & d
\end{pmatrix}.
Step 5: Equate Corresponding Entries
a^2 = a.
This implies
a \in \{0,\,1\}
(since
-1
does not satisfy
(-1)^2 = -1
).
d^2 = d.
Similarly,
d \in \{0,\,1\}.
ab + bd = b.
Factor out
b
to get
b(a + d - 1) = 0.
So either
b = 0
or
a + d = 1.
Step 6: Count the Possible Matrices
Based on the two main cases for
b(a + d - 1) = 0:
Case 1: b \neq 0
If
b \neq 0,
then
a + d = 1.
Since
a, d \in \{0,1\},
the pairs
(a, d)
that sum to 1 are
(0,1)
and
(1,0).
For each pair,
b
can be either
-1
or
1
(assuming nonzero solutions). Hence there are
2 \text{(pairs)} \times 2 \text{(values for } b)
= 4
matrices in this case.
Case 2: b = 0
If
b = 0,
there is no additional restriction on
a + d.
We only require
a \in \{0,1\}
and
d \in \{0,1\}.
Thus there are
2 \text{(choices for } a) \times 2 \text{(choices for } d)
= 4
possibilities in this case.
Step 7: Combine All Possibilities
Summing the counts from both cases:
4 \text{ (Case 1)} + 4 \text{ (Case 2)} = 8.
Therefore, there are
8
such matrices
A
that satisfy
(I - A)^3 = I - A^3
with
a, b, d \in \{-1,0,1\}.
Final Answer
The number of elements in the set is
8.
