• $\begin{array}{l}\mathrm{y=matrix([[1,0,0],[0,2,0],[0,0,3]])*6}\\ \mathrm{>>>}y=\left[\begin{array}{ccc}6& 0& 0\\ 0& 12& 0\\ 0& 0& 18\end{array}\right]\end{array}$

• $\begin{array}{l}\mathrm{m=matrix([[1,2,3],[4,5,6],[7,8,9]])}\\ \det \left(m\right)\\ \mathrm{>>>}0\end{array}$

• $\begin{array}{l}\mathrm{A=matrix([[1,2],[3,5]])}\\ \mathrm{rank(A)}\\ \mathrm{>>>}2\end{array}$

• $\begin{array}{l}\mathrm{m=matrix([[1,2,3],[4,5,6],[7,18,9]])}\\ \mathrm{transpose(m)}\\ \mathrm{>>>}\left[\begin{array}{ccc}1& 4& 7\\ 2& 5& 18\\ 3& 6& 9\end{array}\right]\end{array}$

• $\begin{array}{l}\mathrm{m=matrix([[1,2,3],[4,5,6],[7,18,9]])}\\ \mathrm{adjugate(m)}\\ \mathrm{>>>}\left[\begin{array}{ccc}-63& 36& -3\\ 6& -12& 6\\ 37& -4& -3\end{array}\right]\end{array}$

• $\begin{array}{l}\mathrm{m=matrix([[1,2,3],[4,5,6],[7,18,9]])}\\ \mathrm{inv(m)}\\ \mathrm{>>>}\left[\begin{array}{ccc}-1.05& 0.6& -0.05\\ 0.1& -0.2& 0.1\\ 0.616666666666667& -0.0666666666666667& -0.05\end{array}\right]\end{array}$

• $\begin{array}{l}\mathrm{A=matrix([[3,-2,4,-2],[5,3,-3,-2],[5,-2,2,-2],[5,-2,-3,3]])}\\ \mathrm{EigVal}\left(A\right)\\ \mathrm{>>>}\{-2:1,3:1,5:2\}\end{array}$

Matrix A has eigenvalues -2, 3, and 5, and that the eigenvalues -2 and 3 have algebraic multiplicity 1 and that the eigenvalue 5 has algebraic multiplicity 2.

• $\begin{array}{l}\mathrm{A=matrix([[3,-2,4,-2],[5,3,-3,-2],[5,-2,2,-2],[5,-2,-3,3]])}\\ \mathrm{EigVec}\left(A\right)\\ \mathrm{>>>}\left[\begin{array}{c}(-2,1,\left[\left[\begin{array}{c}0\\ 1\\ 1\\ 1\end{array}\right]\right]),(3,1,\left[\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right]\right]),(5,2,\left[\begin{array}{c}\left[\begin{array}{c}1\\ 1\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ -1\\ 0\\ 1\end{array}\right]\end{array}\right])\end{array}\right]\end{array}$

• $\begin{array}{l}\mathrm{v1=matrix([[1,2,3]])}\\ \mathrm{v2=matrix([[4,5,6]])}\\ \mathrm{dot}(\mathrm{v1},\mathrm{v2})\\ \mathrm{>>>}32\end{array}$

• $\begin{array}{l}\mathrm{v1=matrix([[1,2,3]])}\\ \mathrm{v2=matrix([[4,5,6]])}\\ \mathrm{cross}(\mathrm{v1},\mathrm{v2})\\ \mathrm{>>>}\left[\begin{array}{ccc}-3& 6& -3\end{array}\right]\end{array}$

• $\begin{array}{l}\mathrm{A=matrix([[4,3],[1.5,1]])}\\ L,U=\mathrm{LU}\left(A\right)\\ \mathrm{>>>}L=\left[\begin{array}{cc}1& 0\\ 0.375& 1\end{array}\right]\\ \mathrm{>>>}U=\left[\begin{array}{cc}4& 3\\ 0& -0.125\end{array}\right]\end{array}$

• $\begin{array}{l}\mathrm{A=matrix([[1,1,1],[1,1,3],[2,3,4]])}\\ Q,R=\mathrm{QR}\left(A\right)\\ \mathrm{>>>}Q=\left[\begin{array}{ccc}0.408248290463863& -0.577350269189626& -0.707106781186548\\ 0.408248290463863& -0.577350269189626& 0.707106781186548\\ 0.816496580927726& 0.577350269189626& 0\end{array}\right]\\ \mathrm{>>>}R=\left[\begin{array}{ccc}2.44948974278318& 3.2659863237109& 4.89897948556636\\ 0& 0.577350269189626& 0\\ 0& 0& 1.4142135623731\end{array}\right]\end{array}$

• $\begin{array}{l}\mathrm{A=matrix([[1,2,0],[0,3,0],[2,-4,2]])}\\ P,D=\mathrm{Diag}\left(A\right)\\ \mathrm{>>>}P=\left[\begin{array}{ccc}-1& 0& -1\\ 0& 0& -1\\ 2& 1& 2\end{array}\right]\\ \mathrm{>>>}D=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right]\end{array}$