GATE Civil 2025 Set 1 | Question: 1

Question

​​​​​Suppose $\lambda$ is an eigenvalue of matrix $A$ and $x$ is the corresponding eigenvector. Let $x$ also be an eigenvector of the matrix $B=A-2 I$, where $I$ is the identity matrix. Then, the eigenvalue of $B$ corresponding to the eigenvector $x$ is equal to

• A. $\lambda$
• B. $\lambda+2$
• C. $2 \lambda$
• D. $\lambda-2$

Answer 1

d

Answer 2

Important Rule :
If $x$ is an eigenvector of $A$ with eigenvalue $\lambda$, then $Ax = \lambda x$.

 For any matrix $B = f(A)$, the eigenvalue is $f(\lambda)$ while the eigenvector $x$ remains the same.

Given $B = A - 2I$. Multiplying by eigenvector $x$:
\[ Bx = (A - 2I)x \]
\[ Bx = Ax - 2Ix \]
Since $Ax = \lambda x$ and $Ix = x$:
\[ Bx = \lambda x - 2x \]
\[ Bx = (\lambda - 2)x \]
By the definition $Bx = \lambda_B x$, the eigenvalue of $B$ is $(\lambda - 2)$.

$\lambda - 2$ (Option D) is correct