1. Justin Gilmore Problem Set 1- #15 15.If r is the rank and d is the determinant of the matrix what is r-d? In order to solve the problem, we must first row reduce to Reduce Echelon Form. -4 R1 + R2

2. -7 R1 + R3 -2R2 + R3 To find the rank, r, of the matrix, simply count the number of linearly independent variables. In this case, r = 2.

3. To find the determinant, d, of a 3x3 matrix, use the formula a(ei – fh) – b(di – fg) + c(dh – eg) Set a,b,c as the first column with a = 1, b = 0, and c = 0. Then, 1((-3*0) – (-6*0) – 0 + 0 = 1(0 – 0) = 1*0 = 0 Since, r=2 and d=0, Then r-d= 2.