Q.If matrix $A = [a_{ij}]_{2 \times 2}$, where $a_{ij} = 1$ if $i \neq j$ and $a_{ij} = 0$ if $i = j$, then $A^2$ is equal to
(A) $I$
(B) $A$
(C) $O$
(D) None of these

The key idea is to first construct the $2 \times 2$ matrix $A$ from the given rule, then square it.

Step 1: Build $A$

For $i=j$ (diagonal entries), $a_{ij}=0$. For $i \neq j$ (off-diagonal entries), $a_{ij}=1$.

Thus

Step 2: Compute $A^2$

Multiply $A$ by itself: …

Let’s understand what’s happening. The definition says $a_{ij} = 1$ when $i \neq j$ (off-diagonal entries) and $a_{ij} = 0$ when $i = j$ (diagonal entries). For a $2 \times 2$ matrix, that means:

• $a_{11} = 0$ (since $i = j$)
• $a_{12} = 1$ (since $i \neq j$)
• $a_{21} = 1$ (since $i \neq j$)
• $a_{22} = 0$ (since $i = j$)

So the matrix is:

This is a well-known matrix — it’s the exchange matrix or the flip matrix. When you multiply it by itself, you’re essentially swapping rows/columns twice, which should bring you back to the original. Let’s verify.

1. Write down the matrix explicitly.

   $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.

2. Compute $A^2$ by matrix multiplication.

   Recall: $(A^2)_{ij} = \sum_{k=1}^2 a_{ik} a_{kj}$.

   • For $(1,1)$: $a_{11}a_{11} + a_{12}a_{21} = (0)(0) + (1)(1) = 1$
   • For $(1,2)$: $a_{11}a_{12} + a_{12}a_{22} = (0)(1) + (1)(0) = 0$
   • For $(2,1)$: $a_{21}a_{11} + a_{22}a_{21} = (1)(0) + (0)(1) = 0$
   • For $(2,2)$: $a_{21}a_{12} + a_{22}a_{22} = (1)(1) + (0)(0) = 1$

   So $A^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I$.

3. Interpret the result. …

Method: Build a matrix from its entry rule, then operate

Use this when a matrix is described by a formula for $a_{ij}$ (its general element) rather than written out — you must first construct it, then perform the requested operation.

Steps

Step 1: Translate the entry rule into actual positions.

Go through every position $(i,j)$ and apply the stated condition. For a $2\times2$ matrix the diagonal positions are $(1,1)$ and $(2,2)$ (where $i=j$) and the off-diagonal positions are $(1,2)$ and $(2,1)$ (where $i\neq j$). Fill in each value from the rule.

Step 2: Write the matrix explicitly. …

Common Mistakes

Mistake 1: Misreading the $i=j$ / $i\neq j$ conditions and swapping which entries are $0$ and which are $1$.

Why it's wrong: it builds the wrong matrix from the start, so every later step is wrong. Correct approach: carefully mark diagonal positions ($i=j$) versus off-diagonal ($i\neq j$) before filling values.

Mistake 2: Assuming a zero diagonal forces $A^2 = O$. …