Appeared in: 2024 (6 sets)
Find the matrix $A^2$, where $A = [a_{ij}]$ is a $2 \times 2$ matrix whose elements are given by $a_{ij} = \text{maximum} (i, j) - \text{minimum} (i, j) :$ (A) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ (B) $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ (C) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ (D) $\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$
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Official Evaluation Steps
Step 1 (Definition of Matrix Elements):
The matrix $A = [a_{ij}]$ is a $2 \times 2$ matrix where $a_{ij} = \max(i, j) - \min(i, j)$. We calculate the elements for $i, j \in \{1, 2\}$:
- $a_{11} = \max(1, 1) - \min(1, 1) = 1 - 1 = 0$
- $a_{12} = \max(1, 2) - \min(1, 2) = 2 - 1 = 1$
- $a_{21} = \max(2, 1) - \min(2, 1) = 2 - 1 = 1$
- $a_{22} = \max(2, 2) - \min(2, 2) = 2 - 2 = 0$
Thus, the matrix $A$ is:
$$A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$
Step 2 (Step-by-Step Working):
To find $A^2$, we perform matrix multiplication $A \times A$:
$$A^2 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$Calculating the product:
$$A^2 = \begin{bmatrix} (0 \times 0 + 1 \times 1) & (0 \times 1 + 1 \times 0) \\ (1 \times 0 + 0 \times 1) & (1 \times 1 + 0 \times 0) \end{bmatrix}$$ $$A^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$
Step 3 (Final Answer):
The resulting matrix is the Identity matrix $I_2$. Comparing this with the given options:
Correct Option: (C) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$