(************   Differential  Equations  *************)

x' = -10 x + 10 y
y' = -x  z  + 60 x - y 
z' = x  y  - (8/3) z

odeSystem={
x'[t] == -10 x[t] + 10 y[t],
y'[t] == -x[t]  z[t]  + 60 x[t] - y[t] ,
z'[t] == x[t] y[t]  - (8/3) z[t]
};

initialCondition={
x[0]==1,
y[0]==1,
z[0]==1
};

IVP=Flatten[{odeSystem,initialCondition}];IVP

sol2=NDSolve[IVP, {x,y,z},{t,5}]

ParametricPlot3D[
{x[t],y[t],z[t]}/.sol2,{t,0,5}]

ParametricPlot3D[
{x[t],y[t],z[t]}/.sol2,{t,0,3.3},
PlotPoints->1000,
PlotRange->All,AxesLabel->{"x","y","z"}]

(******  Minimization  *********)

min  y + x^2   subject  to  { x + y = 3.5, -x + 2 <= 0, -y + 1 <= 0}

function[x_,y_]:=y + x^2

constraints={ 
x + y == 3.5,
 -x + 2 <= 0,
 -y + 1 <= 0}

problem=Flatten[{function[x,y] ,constraints}]

minSol=Minimize[ problem,{x,y}]

min3DPoint=
Point[{
x/.minSol[[2,1]],
y/.minSol[[2,2]],
minSol[[1]]
}];
g3D=Graphics3D[{PointSize[0.03],Red,min3DPoint}];

p3D=Plot3D[function[x,y],
{x,-2,4},
{y,-1,3}];

Show[p3D,g3D]

minPoint=
Point[{
x/.minSol[[2,1]],
y/.minSol[[2,2]]
}];
g2D=Graphics[
{PointSize[Large],Red,minPoint}];

rp=RegionPlot[
constraints, 
{x,-2,4},
{y,-1,3}];

Show[rp,g2D]

(*******   Linear  Algebra  ********)

Define  the matrix  (({{1, a, 6, 5}, {-1, 2, 4, 2}, {5 b, 6, b, 4}, {-1, -2, 2, 4}})) dependent on two  parameters "a"  and  "b".
Find   the  values  of  "a"  and  "b"  for  which  the  matrix  is  singular .

mat[a_ , b_]:= {
{1,  a,  6,5},
{-1,2,4,2},
{5b,6,b,4},
{-1,-2,2,4}
}; 
MatrixForm[mat[a,b]]

(* Det[mat[a,b]]/.a->6  *)

Det[mat[a,b]]/.b->6

Det[mat[a,b]]/.{a->6,b->6}

(* Solve[Det[mat[a,b]]==0, a] *)

Solve[Det[mat[a,b]]==0, b]

Solve[Det[mat[a,b]]==0,b]/.a->6

Plot[  Det[mat[-a,a]], {a, -41,50}]

(* Plot[  Det[mat[-b,b]], {b, -15,30}] *)

Plot3D[  Det[mat[a,b]],{a,-41,50}, {b, -15,30}]

Solve[Det[mat[a,b]]==0, {a,b}]

Reduce[Det[mat[a,b]]==0, {a,b}]