Let A be a 3 3 upper triangular matrix whose entries are chosen randomly with replacement from the set 0, 1 . The probability that A is an invertible matrix is :

Step-by-step solution

A 3 3 upper triangular matrix has 6 independent entries ( 3 on the principal diagonal and 3 strictly above it). The total number of such matrices that can be formed using the entries from 0, 1 is 2^6 = 64 . For the matrix A to be invertible, its determinant must be non-zero. The determinant of an upper triangular matrix is the product of its principal diagonal elements. Since the available entries are only 0 and 1 , the product of the diagonal elements is non-zero if and only if all three diagonal elements are exac