- (15 points) For the matrices below, obtain by hand (1) \(A + C\), (2) \(A - C\), (3) \(B'A\), (4) \(AC'\), (5) \(C'A\).
\[
A = \begin{bmatrix}
2 & 1 \\
3 & 5 \\
5 & 7 \\
4 & 8
\end{bmatrix}, \qquad
B = \begin{bmatrix}
6 \\
9 \\
3 \\
1
\end{bmatrix}, \qquad
C = \begin{bmatrix}
3 & 8 \\
8 & 6 \\
5 & 1 \\
2 & 4
\end{bmatrix}
\]
- (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 \(B^{-1}B\).
- (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).
- \(Y_i = \beta_1 X_{i1} + \beta_2 X_{i2} + \beta_3 X^2_{i1} + \varepsilon\)
- \(\sqrt{Y_i} = \beta_0 + \beta_1 X_{i1} + \beta_2 \log X_{i2} + \varepsilon\)