Question

If an eigenvalue of A is \[\lambda \], then the corresponding eigenvalue of \[{A^{ - 1}}\] is.
A.\[ - \lambda \]
B.\[\dfrac{1}{\lambda }\]
C.\[\lambda \]
D.\[\dfrac{{\overline \lambda }}{{{\lambda ^2}}}\]

Answer 1

Hint: Here in this question, we have to find the eigenvalue of the inverse of A using a given eigenvalue of A. This can be solved by multiplying an eigenvector and simplifying using the common transposition method and by using basic arithmetic operation we get the required solution.

Complete step-by-step answer:
Eigenvector is a non-vector in which when a given square matrix is multiplied, it is equal to a scalar multiple of that vector. Let us suppose that A is an \[n \times n\] ordered square matrix, and if \[x\] be a non-zero vector, then the product of matrix A, and vector \[x\] is defined as the product of a scalar quantity \[\lambda \] and the given vector, such that:
\[Ax = \lambda x\]
Where, x is an eigenvector and \[\lambda \] be the scalar quantity that is termed as eigenvalue associated with matrix A.
Eigenvalues are generally associated with eigenvectors in Linear algebra. Eigenvalues are the particular set of scalar values related to linear equations or matrix equations.
In the equation,
\[Ax = \lambda x\]
Where, \[\lambda \] be the eigenvalue of matrix A
Consider the question:
Given, A be an any matrix whose eigenvalue is \[\lambda \] and \[x\] be a corresponding eigenvector, then
\[ \Rightarrow \,\,Ax = \lambda x\]
Divide both side by A, then we have
\[ \Rightarrow \,\,x = \dfrac{1}{A}\lambda x\]
It can also be written as
\[ \Rightarrow \,\,x = {A^{ - 1}}\lambda x\]
or
\[ \Rightarrow \,\,x = \lambda \left( {{A^{ - 1}}x} \right)\]
Now, divide \[\lambda \] on both side, then
\[ \Rightarrow \,\,\dfrac{1}{\lambda }x = {A^{ - 1}}x\]
or
\[ \Rightarrow \,\,{A^{ - 1}}x = \dfrac{1}{\lambda }x\]
Hence, \[\dfrac{1}{\lambda }\] is the eigenvalue of \[{A^{ - 1}}\] and \[x\] is the corresponding eigenvector.
Therefore, Option (B) is correct.
So, the correct answer is “Option B”.

Note: The eigenvectors are connected with the set of linear equations. The other names for the eigenvector of a matrix are latent vector, proper vector, or characteristic vector. Eigenvectors are also useful in solving differential equations and many other applications related to them.