MATH 1ZC3 Study Guide - Midterm Guide: Diagonal Matrix, Symmetric Matrix, Transpose
1 Mar 2013
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Homogeneous systems if all equations are homogeneous, matrix must be consistent: all have the trivial solution (x1 = 0, x2 = 0, , they may have additional solutions as well in this case, they have infinitely many solutions. If system has more unknowns than equations, it has infinitely many solutions thus has a non-trivial solution. If a homogeneous linear system has n unknowns, and r non-zero rows, it has n-r free variables. The trace of a is undefined if a is not square. The trace is the sum of entries on the principal diagonal tr a = tr(a)t tr(a+b) = tr(a) + tr(b) tr(ca) = c tr(a) tr(ab) = tr(ba) Transposes (at)t= a (a+b)t=at + bt (ka)t = kat (ab)t= btat (at) -1 = (a-1)t. Ata and aat are symmetric, and if a is invertible, they are also invertible. Inverses (ab)-1 = a-1b-1 (an)-1 = a-n = (a-1)n (a-1)-1 = a (ka)-1 = k-1a-1.
