Assignemnt 6

  1. (15 points) For the matrices below, obtain by hand (1) \(A + C\), (2) \(A - C\), (3) \(B'A\), (4) \(AC'\), (5) \(C'A\).

\[ A = \begin{bmatrix} 2 & 1 \\ 3 & 5 \\ 5 & 7 \\ 4 & 8 \end{bmatrix}, \qquad B = \begin{bmatrix} 6 \\ 9 \\ 3 \\ 1 \end{bmatrix}, \qquad C = \begin{bmatrix} 3 & 8 \\ 8 & 6 \\ 5 & 1 \\ 2 & 4 \end{bmatrix} \]

  1. (10 points) Find the inverse by hand of each of the following matrices

\[ A = \begin{bmatrix} 2 & 4 \\ 3 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 4 & 3 \\ 6 & 5 \end{bmatrix} \]

Check that these are correct inverse matrices by calculating \(AA^{-1}\) and \(B^{-1}B\).

  1. (10 points) Set up the \(X\) matrix and \(\boldsymbol\beta\) for each of the following regression models (that is, write the model as \(Y = X \boldsymbol\beta + \varepsilon\) and write the vector \(Y\) and \(X\) explicitly).
  1. \(Y_i = \beta_1 X_{i1} + \beta_2 X_{i2} + \beta_3 X^2_{i1} + \varepsilon\)
  2. \(\sqrt{Y_i} = \beta_0 + \beta_1 X_{i1} + \beta_2 \log X_{i2} + \varepsilon\)