Question

The number of elements in the set \[ \left\{ A = \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} : a, b, d \in \{-1, 0, 1\} \text{ and } (I - A)^3 = I - A^3 \right\}, \] where \( I \) is the \( 2 \times 2 \) identity matrix, is _________.

Hint:

Remember that for any square matrix \( A \), if \( (I-A)^n = I-A^n \) for \( n>1 \), it generally implies properties similar to \( A^2=A \). Expanding and comparing coefficients is the safest method.

Answer 1

Correct Answer: 8

Step 1: Understanding the Concept:
The condition \((I - A)^3 = I - A^3\) relates powers of the matrix \( A \). By expanding the binomial power of matrices (since \( I \) and \( A \) commute), we can simplify the equation to find specific properties that the elements \( a, b, \) and \( d \) must satisfy.
Step 2: Key Formula or Approach:
Expand \((I - A)^3\): \[ (I - A)^3 = I^3 - 3I^2A + 3IA^2 - A^3 = I - 3A + 3A^2 - A^3 \]
Given \((I - A)^3 = I - A^3\), we substitute: \[ I - 3A + 3A^2 - A^3 = I - A^3 \implies 3A^2 - 3A = O \implies A^2 = A \]
This means matrix \( A \) must be idempotent.
Step 3: Detailed Explanation:
Given \( A = \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \), calculate \( A^2 \): \[ A^2 = \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} = \begin{pmatrix} a^2 & ab + bd \\ 0 & d^2 \end{pmatrix} \]
For \( A^2 = A \), we must have: 1. \( a^2 = a \implies a(a-1) = 0 \implies a \in \{0, 1\} \)
2. \( d^2 = d \implies d(d-1) = 0 \implies d \in \{0, 1\} \)
3. \( ab + bd = b \implies b(a + d - 1) = 0 \)
Now, we analyze the cases for \( b \in \{-1, 0, 1\} \):
- Case 1: \( b = 0 \)
The equation \( 0(a+d-1)=0 \) is always true. Since \( a \in \{0, 1\} \) and \( d \in \{0, 1\} \), there are \( 2 \times 2 = 4 \) such matrices.
- Case 2: \( b \neq 0 \) (\( b \in \{-1, 1\} \))
The equation \( b(a+d-1)=0 \) implies \( a + d = 1 \).
- If \( a = 1 \), then \( d = 0 \). This gives \( b \in \{-1, 1\} \) (2 matrices).
- If \( a = 0 \), then \( d = 1 \). This gives \( b \in \{-1, 1\} \) (2 matrices).
Total number of matrices = \( 4 \text{ (from Case 1)} + 2 + 2 = 8 \).
Step 4: Final Answer:
The number of elements in the set is 8.