• I. Let V = M_{2 x 2}(R) denote the set of all 2 x 2 matrices with real entries. Note that V is a vector space under the addition and scalar product operations for matrices.
  • A) Let B be the set of four matrices B = {E₁₁,E₁₂, E₂₁,E₂₂}, where E_ij has a 1 in row i and column j and zeroes in all other entries. Show that B is a basis for V.
  • B) Let A be a fixed A matrix. Show that the mapping T: V -> V defined by T(X) = AX (matrix product) is linear.
  • C) If A = a E₁₁ + bE₁₂ + cE₂₁ + dE₂₂, find the matrix of T with respect to B,B.
• II. Let A be an n x n matrix.
  • A) Show that A is invertible if and only if the system Ax = E_i has a unique solution for each standard basis vector E_i in Rⁿ.
  • B) Explain why you can solve all of these systems simultaneously by making an n x 2n matrix M = [A|I] (where the second block is the n x n identity matrix), and reducing the matrix M to echelon form.
  • C) When you do this, what matrix appears in the right-hand block of the echelon form matrix?
  • D) Compute the inverses of the matrices in problem 7 from Chapter 13 in Smith.
• III. From Smith, Chapter 13: 1 a,b, 6 a,b, 8, 9.

Due: Friday, April 9.