Let $A=[a_{ij}]$ be a square matrix of order $2$ with entries either $0$ or $1$. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is:
Step-by-Step Solution
Key Concept: Total $2\times 2$ matrices with $0/1$ entries: $2^{4}=16.$ Count invertible by $ad-bc\ne 0$, which is equivalent to $ad\ne bc.$
Total: $16.$ Singular: $ad=bc.$
$ad=bc=0$: $(a,d)\in\{(0,0),(0,1),(1,0)\}$ and $(b,c)\in\{(0,0),(0,1),(1,0)\}$: $3\cdot 3=9$ matrices.
$ad=bc=1$: requires $a=d=b=c=1$, $1$ matrix.
Singular total: $10.$ Invertible: $6.$ $P(E)=\dfrac{6}{16}=\dfrac{3}{8}.$
Correct Answer: 3