Consider the 2x3 matrix  A = 1 0 4                                                   0 2 1 over the field Q of rationals. Using elementary row and column operations, or otherwise, find invertible matrices P, Q such that the matrix PAQ is in canonical form for equivalence Let A be an n × n matrix with a column of zeros. Show directly from the Leibniz formula for determinants that det(A) = 0. Now suppose that V has a basis v1, v2, v3 and W a basis w1, w2, and let α : V → W be the linear map corresponding to the matrix A in (b) with respect to these bases. State the values α(v1), α(v2), α(v2). What is the rank ρ(α)?

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