374 - The Work for Examples 1 and 2 in Section 8.6

Example 1 Work

A={{1,-2,2},{-2,1,-2},{2,-2,1}}
Out[]=
{{1,-2,2},{-2,1,-2},{2,-2,1}}
MatrixForm[A]
Out[]//MatrixForm=
1
-2
2
-2
1
-2
2
-2
1
Det[A-m*IdentityMatrix[3]]
Out[]=
5+9m+3
2
m
-
3
m
Simplify[%]
Out[]=
-((-5+m)
2
(1+m)
)
MatrixForm[A+1*IdentityMatrix[3]]
Out[]//MatrixForm=
2
-2
2
-2
2
-2
2
-2
2
MatrixForm[RowReduce[%]]
Out[]//MatrixForm=
1
-1
1
0
0
0
0
0
0
MatrixForm[A-5*IdentityMatrix[3]]
Out[]//MatrixForm=
-4
-2
2
-2
-4
-2
2
-2
-4
MatrixForm[RowReduce[%]]
Out[]//MatrixForm=
1
0
-1
0
1
1
0
0
0

Example 2 Work

In[]:=
Clear[A]
A={{3,-18},{2,-9}}
Out[]=
{{3,-18},{2,-9}}
MatrixForm[A]
Out[]//MatrixForm=

3
-18
2
-9

Eigenvalues[A]
Out[]=
{-3,-3}
Eigenvectors[A]
Out[]=
{{3,1},{0,0}}
In[]:=
Solve[(A+3*IdentityMatrix[2]).{{c1},{c2}}{{3},{1}},{c1,c2}]
Solve
:Equations may not give solutions for all "solve" variables.
Out[]=
c2-
1
6
+
c1
3

Thus,withc2=0,c1=
1
2
.