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.
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.