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 an invertible matrix. Then the probability \( P(E) \) is:

• A. \( \frac{3}{16} \)
• B. \( \frac{3}{8} \)
• C. \( \frac{5}{8} \)
• D. \( \frac{1}{8} \)

Hint:

To determine the probability of an event involving matrices, count the total number of possible matrices and the favorable cases (invertible or non-invertible) and calculate the ratio.

Answer 1

The Correct Option is C

A 2x2 matrix is invertible if its determinant is non-zero. The total number of 2x2 matrices with entries 0 or 1 is \( 2^4 = 16 \). To find the number of invertible matrices, we identify and subtract the count of non-invertible matrices (those with a determinant of zero) from the total. The probability \( P(E) \) is determined by the ratio of invertible matrices to the total matrix count.
Final Answer: \( \frac{5}{8} \).