Let A=[a_ij] be a square matrix of order 2 with entries either 0 or 1.

Question

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 invertible. Then the probability $P(E)$ is:

Accepted Answer

Total matrices $=2^4=16$. $A$ is invertible iff $ad-bc\ne0$.

Counting singular matrices: those with $\begin{pmatrix}1&1\0&1\end{pmatrix}$-type pattern give 4 singular cases, and permutations of $\begin{pmatrix}1&0\0&1\end{pmatrix}$-type structures give 2 more --- total singular matrices $=10$, so invertible $=6$.

$$P(E) = \frac{6}{16}=\frac38$$

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