Practice Problems
MTH 2201 2/10/2009
1. Find the eigenvalues and eigen vectors of
i) A="10 −9
4−2#.Solution: (i) λ= 4,4; X=t"3
2#
(ii) A=
3 4 −1
−1−2 1
3 9 0
Solution: (i) λ1= 2,2; X=t
−1
1
3
;λ2=−3; X=t
−1
1
−2
.
2. Find the eigen values and eigen vectors of Aand of the stated power of A.
(i) A=
−1−2−2
1 2 1
−1−1 0
;A25
Solution: λ1=−1; X=t
2
−1
1
;λ2= 1; X=t
0
−1
1
+s
1
−1
0
.
The eigen values of A25 are λ= (−1)25 =−1 and λ2= (1)25 = 1.
3. For what value(s) of xif any does the matrix A=
3 0 0
0x2
0 2 x
,has atleast
one repeated eigenvalue. (solution: x= 1 or x= 5.)
4. Let Abe a (2 ×2) matrix such that A2=I. For any x∈IR2, if x+Axand
x−Axare eigenvectors of Afind the corresponding eigenvalue.
5. Prove that if Ais a square matrix then Aand AThave the same characteristic
polynomial.
6. Let A="2 0
2 3 #. Show that Aand ATdo not have the same eigen spaces.
7. If λis an eigen value of Aand Xis the corresponding eigenvector, then prove
that λ−sis an eigen value of A−sI for any scalar sand Xis the
corresponding eigenvector.
8. Let A="2 0
2 3 #and B1, B2, B3be the matrices obtained by the elementary
row operations R2→R2−R1,R2↔R1and R2→(−2)R2respectively on A.
Find the eigen values of A, B1, B2and B3.
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