Suppose you have a $2 \times 2$ matrix, $a$ that can only take the entries 1 or 2. Then, suppose you have a 2-vector $b$ that takes the values 1 or 2 as well. Looking at $Ax = b,$ the probability space has 64 elements.
What is the probability the system has a unique solution?
Alright, there's gotta be an easier way to do this than listing all possible matrices and then finding if it has a unique solution. Can someone please guide me in the right direction? I seriously have no idea.
I believe the $b$ vector is a red herring. It doesn't matter. In order for the matrix $A$ to be invertible, it's determinant must be nonzero. The probability that this determinant is nonzero is one minus the probability that it is zero. The determinant is $$ \det A = A_{11}A_{22} - A_{12} A_{21}\, . $$ There are $4^2 = 16$ possibilities for this matrix. Can you count how many lead to a zero determinant, and thus find the probability that the determinant is not zero?