For a n-by-n square matrix M, list five different conditions that each one of them can determine that M is invertible.

1. The determinant of M is 0. 2. The rank of M is n. 3. The linear equations MX = 0 (where X is a n-by-1 vector) have and only have one solution, which is (0,0,0,.......,0). 4. The column vectors (or row vectors) are linearly independent. 5. Not all of the eigenvalues of M are 0.

Given f(x) = a for x = 0 and f(x) = [exp(-2x) - 1] / x otherwise, and f(x) is a continuous function, calculate a.

By the definition of continuity, f(x) must have limit at x = 0 and the limit must exactly be f(0). To calculate the limit value at x = 0, using L' Hospital Rule is -2 so a is equal to -2.

There are two containers. The first one contains 4 red balls and 5 white balls while the second one contains 3 red balls and 8 white balls. Now take a ball randomly from all these balls. Given that it is red, what is the probability that it comes from the first container?

Solution: ( ( 9 / 20 ) * ( 4 / 9 ) ) / ( ( 9 / 20 ) * ( 4 / 9 ) + ( 11 / 20 ) * ( 3 / 11 ) ) = 4 / 7