Math 2243 Midterm II, 2:30 Page 7 Name: 8. (10 pts) Which of the following prope
ID: 2885918 • Letter: M
Question
Math 2243 Midterm II, 2:30 Page 7 Name: 8. (10 pts) Which of the following properties are equivalent to an n xn matzix 4 beting matrix A being invertible? (A) A is row equivalent to an identity matrix I (B) AT 0 has only the trivial solution. (C) At has a unique solution for any (having the same dimension as AT). (D) Ax = b is consistent for any b (having the same dimension as A (E) A is nonsingular (F) The solution space of A 0 has dimension 0. 9. (10 pts) Suppose that () y"+ay+ by 0, where a and b are real numbers, and let V denote the set of all solutions. Which of the following are correct? (A) If yi, y2, ys are solutions of (), then so is 8y] - y2 3ys (B) If y, V2, ys are solutions of (), then they might span V (C) If 3v1 32 V3 are solutions of (+), then they might be linearly independent. (D) et and e might both be solutions of (. (E) e and sin r might both be solutions of (). (F) e" and 1 might both be solutions of (*). Point total for page:Explanation / Answer
8) A square N x N matrix is invertible, if its discriminant is non-zero. This is because in order to find the inverse of a matrix, we need to divide the conjugate of its adjoint matrix by the determinant value of A. Another way to say this condition is that the matrix A is non singular. So our option is E
9) The generalsolution to the given differential equation is of the form
y = C1 exp(r1x) + C2 exp(r2x)
where exp() denotes the exponential functions, and the r1 and r2 can be determined as per corresponding values of a and b. C1 and C2 are any arbitrary constants.
Hence, if r1 = 1 and C2 = 0, we get one solution as ex. And if r1 = -1 and C2 = 0, we get another solution as e-x. Hence option (D) is correct.