$\mathrm{with}\left(\mathrm{LinearAlgebra}\right)\:$

$M≔\mathrm{RandomMatrix}\left(5\right)$

$M≔\left[\begin{array}{ccccc}\mathrm{-81}& \mathrm{-98}& \mathrm{-76}& \mathrm{-4}& 29\\ \mathrm{-38}& \mathrm{-77}& \mathrm{-72}& 27& 44\\ \mathrm{-18}& 57& \mathrm{-2}& 8& 92\\ 87& 27& \mathrm{-32}& 69& \mathrm{-31}\\ 33& \mathrm{-93}& \mathrm{-74}& 99& 67\end{array}\right]$

$\mathrm{SubMatrix}\left(M\,\left[2..5\right]\,\left[2..3\,1\right]\right)$

$\left[\begin{array}{ccc}\mathrm{-77}& \mathrm{-72}& \mathrm{-38}\\ 57& \mathrm{-2}& \mathrm{-18}\\ 27& \mathrm{-32}& 87\\ \mathrm{-93}& \mathrm{-74}& 33\end{array}\right]$

$\mathrm{SylvesterMatrix}\left(x^2+3x\,2x\right)$

$\left[\begin{array}{ccc}1& 3& 0\\ 2& 0& 0\\ 0& 2& 0\end{array}\right]$

$\mathrm{Eigenvectors}\left(\mathrm{Matrix}\left(\left[\left[4\,-1\,6\right]\,\left[2\,1\,6\right]\,\left[2\,-1\,8\right]\right]\right)\right)$

$\left[\begin{array}{c}9\\ 2\\ 2\end{array}\right],\left[\begin{array}{ccc}1& \mathrm{-3}& \frac{1}{2}\\ 1& 0& 1\\ 1& 1& 0\end{array}\right]$

$M≔\mathrm{Matrix}\left(\left[\left[\frac{\mathrm{sqrt}\left(10\right)\cdot 3}{10}\,-\frac{\mathrm{sqrt}\left(10\right)}{10}\right]\,\left[\frac{\mathrm{sqrt}\left(10\right)}{10}\,\frac{\mathrm{sqrt}\left(10\right)\cdot 3}{10}\right]\right]\right)$

$M≔\left[\begin{array}{cc}\frac{3\sqrt{10}}{10}& -\frac{\sqrt{10}}{10}\\ \frac{\sqrt{10}}{10}& \frac{3\sqrt{10}}{10}\end{array}\right]$

$\mathrm{IsOrthogonal}\left(M\right)$

$\mathrm{true}$

$M≔\mathrm{Matrix}\left(\left[\left[1\,1\,3\,-1\right]\,\left[1\,1\,1\,1\right]\,\left[1\,-2\,1\,-1\right]\,\left[4\,1\,8\,-1\right]\right]\right)$

$M≔\left[\begin{array}{cccc}1& 1& 3& \mathrm{-1}\\ 1& 1& 1& 1\\ 1& \mathrm{-2}& 1& \mathrm{-1}\\ 4& 1& 8& \mathrm{-1}\end{array}\right]$

$v≔\mathrm{Vector}\left(\left[0\,1\,1\,0\right]\right)$

$v≔\left[\begin{array}{c}0\\ 1\\ 1\\ 0\end{array}\right]$

$\mathrm{LinearSolve}\left(M\,v\right)$

$\left[\begin{array}{c}\frac{25}{6}\\ \frac{4}{3}\\ -\frac{5}{2}\\ \mathrm{-2}\end{array}\right]$

$u≔\mathrm{Vector}\left(\left[1\,3\right]\right)$

$u≔\left[\begin{array}{c}1\\ 3\end{array}\right]$

$v≔\mathrm{Vector}\left(\left[5\,7\right]\right)$

$v≔\left[\begin{array}{c}5\\ 7\end{array}\right]$

$u+v$

$\left[\begin{array}{c}6\\ 10\end{array}\right]$

$u·v$

$26$

$A≔\mathrm{Matrix}\left(\left[\left[1\,3\right]\,\left[5\,7\right]\right]\right)$

$A≔\left[\begin{array}{cc}1& 3\\ 5& 7\end{array}\right]$

$B≔\mathrm{Matrix}\left(\left[\left[1\,1\right]\,\left[1\,1\right]\right]\right)$

$B≔\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$

$A+2B$

$\left[\begin{array}{cc}3& 5\\ 7& 9\end{array}\right]$

$A·B$

$\left[\begin{array}{cc}4& 4\\ 12& 12\end{array}\right]$

$A·u$

$\left[\begin{array}{c}10\\ 26\end{array}\right]$

$A^{-1}$

$\left[\begin{array}{cc}-\frac{7}{8}& \frac{3}{8}\\ \frac{5}{8}& -\frac{1}{8}\end{array}\right]$

$A^{\mathrm{\%T}}$

$\left[\begin{array}{cc}1& 5\\ 3& 7\end{array}\right]$

For more information including:

a complete list of the routines in the LinearAlgebra package

the supported data structures and data types

the different sets of commands based on usage scenario: casual use or programming use

the LinearAlgebra:-Modular subpackage for performing dense linear algebra computations in Z/m.

the LinearAlgebra:-Generic subpackage for computing with generic implementations of algorithms for linear algebra over fields, Euclidean domains, integral domains and rings.

see the Details of the LinearAlgebra Package help page.