Let $A = [a_{ij}]$ be a square matrix of order 2 with elements either 0 or 1. Then the difference between the possible number of singular and non-singular matrices is

4

The correct answer is Option (1) → 4

Total number of $2\times2$ matrices with entries $0$ or $1$

$=2^{4}=16$

Let $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ where each entry is $0$ or $1$

$A$ is singular when $|A|=ad-bc=0$

Possible singular cases

Case 1: Both rows are zero

$\begin{pmatrix}0&0\\0&0\end{pmatrix}$ → $1$ matrix

Case 2: One row zero, other non-zero

Non-zero binary row possibilities: $(1,0),(0,1),(1,1)$ → $3$

Row1 zero → $3$ matrices, Row2 zero → $3$ matrices → $6$ total

Case 3: Both rows equal and non-zero

$(1,0),(0,1),(1,1)$ → $3$ matrices

Total singular matrices

$=1+6+3=10$

Hence non-singular matrices

$=16-10=6$

Difference between number of singular and non-singular matrices

$=10-6=4$

The required difference is $4$.