Problem 1

Consider the matrix

\[ \pmb{A} = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix} \]

  1. Show that \(\pmb{A}\) is positive definite.

  2. Find \(\pmb{A}^{1/2}\).

Problem 2

Let

\[ \pmb{A} = \begin{pmatrix} 1 & 0 & x \\ 0 & 1 & 0 \\ x & 0 & 1 \\ \end{pmatrix} \]

Find \(\frac{\partial \log\vert\pmb{A}\vert}{\partial x}\)

  1. directly

  2. using theorem (4.3.30)

Problem 3

Say \(\pmb{Y} = \begin{pmatrix} Y_1& Y_2 \end{pmatrix}'\) with \(E[Y_1]=E[Y_2]=0\), \(var(Y_1)=1\), \(var(Y_2)=4\) and \(cov(Y_1,Y_2)=-2\). Let \(\pmb{A} = \begin{pmatrix} 2 & -1 \\ 0 & 3 \end{pmatrix}\).

Verify theorem (5.1.10) part i. directly.