Math 340 Fall 2021 Worksheet 31. For a nonsingular matrix A and a nonnegative integer p, show that (Ap)−1= (A−1)p.2. Show that if A is skew-symmetric, then all of the elements on the main diagonal of A are 0.3. Find the inverses of the following matricies:(a) A =1 35 2(b) B =1 22 14. If A−1=3 21 3and B−1=2 53 −2compute (AB)−1.5. Find two 2 × 2 singular matricies whose sum is nonsingular.6. The linear system A2x = b has A is nonsingular and A−1=3 02 1and b =−12. Find x.17. Let f : R27→ R2be defined by f(x) = Ax where A =1 20 11 1. Determine which of the followingvectors are in the range of f:(a) w =1−12(b) w =111(c) w =000(d) w =853(e) w =142(f) w =1−118. Let f : Rn7→ Rmbe a matrix transformation defined by f(u) = Au where A is an m × n matrix. Showthat(a) f(u + v) = f(u) + f(v) for any u, v ∈ Rn.(b) f(cu) = cf (u) for any u ∈ Rnand any real number c.(c) f(cu + dv) = cf(u) + df(v) for any u, v ∈ Rnand real numbers c, d.9. Determine the reduced row echelon form of A =cos(θ) − sin(θ)sin(θ)
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