Problem 1

Let

\[ \pmb{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

Using (4.2.3) verify that \(\left(\pmb{A}^{-1}\right)^{-1}=\pmb{A}\).

Problem 2

Show that for any square \(n\times n\) matrix \(\pmb{A}\) the quadratic form can be written as

\[\pmb{y'Ay}=\pmb{y'}\left(\frac{\pmb{A}+\pmb{A'}}{2}\right)\pmb{y}\]

Problem 3

Let

\[ \pmb{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

Show that \(\vert\pmb{A}^2\vert=\vert\pmb{A}\vert^2\)

Problem 4

Find all the solutions of the system of equations

\[ \begin{aligned} &2x+3y-z =1 \\ &x-y = 1 \end{aligned} \]

  1. directly

  2. using theorem (4.2.16)

Problem 5

Say

\[ \pmb{A} = \begin{pmatrix} \frac12 & -\frac14 \\ -\frac14 & \frac12 \end{pmatrix} \]

Find \(\log(1+\pmb{A})\).