Question

Let A = [aij] 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 :

Answer 1

3/8

Answer 2

The total number of possible 2x2 matrices with entries from {0, 1} is $2^4 = 16$. A matrix is invertible if its determinant is non-zero. For a matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is $ad - bc$. The matrix is non-invertible if $ad - bc = 0$, which means $ad = bc$.

Case 1: $ad = 0$ and $bc = 0$. There are 3 ways for $ad=0$: (0,0), (0,1), (1,0). There are 3 ways for $bc=0$: (0,0), (0,1), (1,0). Number of matrices in this case = $3 \times 3 = 9$.

Case 2: $ad = 1$ and $bc = 1$. There is 1 way for $ad=1$: (1,1). There is 1 way for $bc=1$: (1,1). Number of matrices in this case = $1 \times 1 = 1$.

Total non-invertible matrices = $9 + 1 = 10$. Total invertible matrices = $16 - 10 = 6$. Probability $P(E) = \frac{6}{16} = \frac{3}{8}$.