Exercises

1.: Without calculating
$A^{-1}$
establish wether a given matrix A is invertible. Hint: A is invertible if it can be transformed by elementary row operations into a triangular form with non-zero elements on its diagonal.

$\begin{displaymath} (a)\;\;\;A=\left( \begin{array}{cc} 2 & 5 \ 1 & 4 \end{ar... ... & 3 \ 1 & 1 & 4 & 9 \ 1 & -1 & 8 & 27 \end{array}\right) \end{displaymath}$
2.: Find the inverse $A^{-1}$ of A if $A^{-1}$ exists.

$\begin{displaymath} (a)\;\;\;A=\left( \begin{array}{cc} 1 & 2 \ -3 & -1 \end{... ... 1 & 0 \ 0 & 1 & 2 & 1 \ 0 & 0 & 1 & 2 \end{array}\right) \end{displaymath}$

$\begin{displaymath} (d)\;\;\;A=\left( \begin{array}{ccccc} 2 & 1 & 0 & 0 & 0 \\ ... ... 0 & 0 & 0 & 2 & 1 \ 0 & 0 & 0 & 0 & 2 \end{array}\right) \end{displaymath}$
3.: Given a linear system
$A\cdot x = b$
with a square matrix A find the solution of the system as
$x = A^{-1}b.$

$\begin{displaymath} (a)\;\;\;\left( \begin{array}{cc} 1 & 2 \ 3 & 4 \end{array... ...}\right) = \left( \begin{array}{c} 4 \ 10 \end{array}\right) \end{displaymath}$

$\begin{displaymath} (b)\;\;\;\left( \begin{array}{ccc} 1 & 2 & 1\ 2 & 1 & 0 \\... ...ht) = \left( \begin{array}{c} 0 \ 2 \ 0 \end{array}\right) \end{displaymath}$

$\begin{displaymath} (c)\;\;\;\left( \begin{array}{ccccc} 1 & 1 & -1 & 0 & 0\ -... ... \begin{array}{c} 1 \ 0 \ 1 \ 0 \ 9 \end{array}\right) \end{displaymath}$

Sergey Nikitin 2004-01-28