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:
Show Hint
- \( \frac{3}{16} \)
- \( \frac{3}{8} \)
- \( \frac{5}{8} \)
- \( \frac{1}{8} \)
The Correct Option is C
Solution and Explanation
A 2x2 matrix is invertible if its determinant is non-zero. We count the total number of 2x2 matrices with entries 0 or 1, which is \( 2^4 = 16 \). Then, we count the number of matrices that are not invertible (i.e., their determinant is zero) and subtract that from the total to find the number of invertible matrices. The probability \( P(E) \) is the ratio of invertible matrices to the total number of matrices.
Final Answer: \( \frac{5}{8} \).
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