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Math Made Easy-Tricks on Eigenvalues

Maths Trick-16
This is a trick on eigenvalues with a solved problem from IIT JAM paper.

Trick:- If k is an eigenvalue of A, then kⁿ is an eigen value of Aⁿ , for any positive integer n.
Proof:- Since k is an eigenvalue of A.
For  x ≠ 0,  Ax=kx
       =>   A(Ax)=A(kx)
       =>  A²x = k²x
      => k² is an eigenvalue of A².
We have, A²x=k²x
          => A(A²x)=A(k²x)
         =>  A³x = k³x
     => k³ is an eigenvalue of A³.
In the same way, we can prove that for any n=2,3,4, . . . ; kⁿ is an eigenvalue of Aⁿ.

Observation:- If a square matrix has 0 as its eigenvalue then the matrix is singular.
Justification:- | A-k.I |=0  is the characteristic equation, where k is the eigenvalue of A.
If k=0, then |A|=0.

Problem:- [ IIT-JAM-05' Mathematics ] Let A be a 3×3 matrix with eigenvalues 1,−1 and 3. Then
A.   A² + A is non-singular
B.   A² − A is non-singular
C.   A² + 3A is non-singular
D.   A² − 3A is non-singular
Sol:-  (C)  Eigenvalues of A is 1,−1,3.
So, eigenvalues of A² is 1,1,9
=> eigenvalues of A² + 3A is
 1+3.1, 1+3(−1), 9+3(3)
i.e.  4, −2 and 18.
As no eigenvalue is 0 ,so A² + 3A is non-singular.