Find the matrix $A^2$, where $A = [a_{ij}]$ is a $2 \times 2$ matrix whose elements are given by $a_{ij} = \text{maximum}(i, j) - \text{minimum}(i, j)$
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ ##
Given for matrix $A=[a_{ij}]_{2 \times 2}$
We have, $a_{ij} = \text{maximum}(i, j) - \text{minimum}(i, j)$
$a_{11} = 1 - 1 = 0$
$a_{12} = 2 - 1 = 1$
$a_{21} = 2 - 1 = 1$
$a_{22} = 2 - 2 = 0$
$A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
$A^2 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$