Exercises

Exercises


	Exercises

1.

Calculate determinants of the following matrices.

$\begin{displaymath} \left( \begin{array}{cccc} 2 & 1 & 0 & 0\ 1 & 2 & 1 & 0\... ... 0 & 0 & 1 & 2 & 4\ 0 & 0 & 0 & 1 & 5\ \end{array}\right) \end{displaymath}$

2.

Prove that

$\begin{displaymath} \det \left( \begin{array}{cccc} 1 & 1 & \dots & 1 \ \la... ...^{n-1} \end{array}\right)= \prod_{i>j}(\lambda_i - \lambda_j), \end{displaymath}$

where

$\prod_{i>j}(\lambda_i - \lambda_j)$

denotes the product of numbers

$\begin{displaymath} \{\lambda_i - \lambda_j;\;\;\mbox{ where } i>j\}. \end{displaymath}$

3.

Calculate the determinant of the following nxn matrix as a function of $n.$

$\begin{displaymath} \left( \begin{array}{ccccc} n & 1 & 0 & \dots & 0 \ 1 &... ... &\dots & \vdots \ 0 & 0 & 0 &\dots & 1 \end{array}\right) \end{displaymath}$

4.

Use Cramer's rule to find the solution of the following systems.

a.: $\begin{displaymath} \left\{ \begin{array}{ccccccccc} x_1& + & x_2 & + & x_3 & + ... ...+ & 16 x_2 & + & 27 x_3 & + & 64 x_4 &=& 62 \end{array}\right. \end{displaymath}$
b.: $\begin{displaymath} \left\{ \begin{array}{ccccccccc} x_1 & + & x_2 & + & x_3 & +... ... & + & 16 x_2 & + & 27 x_3 & + & x_4 &=&45 \end{array}\right. \end{displaymath}$

Sergey Nikitin 2004-09-09