Question

Number of 2 × 2 matrices A = {aij}2×2 such that |aij| = |aji|, where |aij| Î {0, 1, 2, 3, 4, 5} is -

Answer 1

216

Answer 2

Let the matrix be $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$.

The elements $a_{ij}$ are from the set $S = \{0, 1, 2, 3, 4, 5\}$.

The condition is $|a_{ij}| = |a_{ji}|$. Since $a_{ij} \in S$, $a_{ij} \ge 0$, so $|a_{ij}| = a_{ij}$.

The condition becomes $a_{ij} = a_{ji}$. This means the matrix must be symmetric: $a_{12} = a_{21}$.

The elements $a_{11}, a_{12}, a_{22}$ can be chosen independently from the set $S$. Once $a_{12}$ is chosen, $a_{21}$ is determined as $a_{21} = a_{12}$.

• Number of choices for $a_{11}$ is $|S| = 6$.
• Number of choices for $a_{12}$ is $|S| = 6$.
• Number of choices for $a_{22}$ is $|S| = 6$.

The total number of such matrices is the product of the number of independent choices: $6 \times 6 \times 6 = 6^3 = 216$.