Question

If a matrix is squared, then

• A. both eigenvalues and eigenvectors must change
• B. both eigenvalues and eigenvectors are retained
• C. eigenvalues get squared but eigenvectors are retained
• D. eigenvalues are retained but eigenvectors change

Hint:

If $A\mathbf{v}=\lambda\mathbf{v}$, then $A^n\mathbf{v}=\lambda^n\mathbf{v}$ — eigenvectors stay the same, eigenvalues raise to the $n$th power.

Answer 1

The Correct Option is C

If a matrix $A$ has an eigenvalue $\lambda$ with corresponding eigenvector $\mathbf{v}$, then by definition:
\[ A\mathbf{v} = \lambda \mathbf{v} \]
Now consider the squared matrix $A^2$:
\[ A^2 \mathbf{v} = A(A\mathbf{v}) = A(\lambda \mathbf{v}) = \lambda (A\mathbf{v}) = \lambda (\lambda \mathbf{v}) = \lambda^2 \mathbf{v} \]
This shows that $\mathbf{v}$ remains an eigenvector of $A^2$, but the corresponding eigenvalue becomes $\lambda^2$.
Therefore:
– Eigenvectors are retained.
– Eigenvalues get squared.
Hence, option (C) is correct.