Ex1

A={{-1,3,2},{0,-2,1},{1,0,-2}}

B={{0,6,0},{7,5,9},{8,6,2}}

i)

Det[A]

Tr[A]

ii)

Transpose[A]//MatrixForm

Inverse[A]//MatrixForm

iii)

Eigenvalues[A]//N

Eigenvectors[A]//N

iv)

A.B//MatrixForm

v)

A^(21)//MatrixForm

(*or*)

Power[A,21]//MatrixForm

Ex2

Solve[{3 x-y==2,x+y+4 z==6,2 x+y+5 z==8},{x,y}]

The system is indefinite with one parametric family of  solutions.

Ex3

A1={{4,2,1},{2,4,1},{1,4,2}}

u={{0,0,1}}//MatrixForm

Det[A1]

(*It is ≠ 0*)

inv=Inverse[A1]

x=inv.Transpose[u]//MatrixForm

Ex4

RSolve[{x[n+2]-4 x[n]==-3 n+2,x[0]==0,x[1]==0},x[n],n]

Ex5

lst1=Table[x y^(1/2),{x,2,3},{y,4,8}]

Ex6

Clear[y]

sol=DSolve[{y'[x]==x*y[x],y[1]==2},y[x],x]

Plot[y[x]/.sol,{x,-2,2}]

Ex7

ClearAll[y]

DSolve[{y''[x]+5 y'[x]+6 y[x]==0,y[0]==2,y'[0]==3},y[x],x]

Plot[(E^(-3 x)) (-7+9 (E^x)),{x,0,2},PlotStyle->{Dashed,Red},PlotLabel->"Solution of y''+5y'+6y=0,y(0)=2,y'(0)=3"]

Ex8

Plot[Which[x<0,x,x<1,0,1≤x,x-1],{x,-Pi,Pi}]