1. Ann x n matrix is called nonsingular if it is the matrix of coefficients of a homogeneous system with a unique solution. Determine whether or not the following matrix is nonsingular. 2. Find all solutions to the following systems. Write your answer in vector form and identify both the particular solution and the solution to the homogeneous system. (a) 2x-y-z+w=4 x+y+z=-1 (b) x-2y-z-2w = 4 2 2 1] 105 14 -3x+6y+z-4w=-6 4x-8y-32-3w=13 3. A quadratic function has the form f(x) = ax+bx+c, where a, b, and c are real numbers. Use linear algebra to find all quadratic functions such that f(1) = 0 and f(3) = 0. 4. Can [3] be written as a linear combination of the vectors [4] and [2 cients a and b such that If it can, find coeffi- If not, explain why. Hint: try and write down a system of equations the coefficients a and bi have to satisfy. 5. Unlike multiplication of numbers, matrix multiplication might not be commutative. That is, for two nxn matrices A and B, it might be the case that A BB.A. Find two 2 x 2 matrices that are not commutative and show that they are not. Hint: don't overthink this. Most of the time, two randomly-selected matrices don't commute. You just have to show that the two you pick don't.