A FEW LINEAR ALGEBRA QUESTIONS

A few questions on Linear Algebra. for example 1)Consider this linear system: x + 3y - 2z = 2, 2x +5y + 1z = 0, -3x - 4y + -5z = 0. The system has a unique solution for x, y, z. Find x. (Hint: it is an integer.) If an equation contains a double minus then treat it as a plus. E.g. - -9 = +9. 2) Consider the following linear system with variables x, y, z and an unspecified constant a: x -3z = -1, 1x + y +2z = -1, -2x + ay +-3z = 6. There is one value of a for which this system is inconsistent, i.e does not have a solution. Find this value of a. (Enter your answer to 4 decimal places.) 3) Consider the 3 x 3 matrix A with the following rows: 1st row: 1 -1 7 2nd row: 1 -2 -4 3rd row: 3 1 1 Find the (2,3)-entry of the inverse of A. (Enter your answer to 4 decimal places.) 4) Consider the function f(x,y) = y ln(1+(xy)2). .Determine the value of a such that the direction of the vector (1,a) is the direction in which f is increasing fastest at the point (1,2). 5) Consider the function f(x,y) = x8y3cos(pi x3y4) where pi denotes the number pi = 3.14159....Evaluate the integral of this function over the rectangular domain specified by 0 =< x =< 1 and 0 =< y =< 1.