GATE Electrical 2026 | Question: 20

Question

Two $n \times n$ matrices $A$ and $B$ have a common eigenvalue $2$, and the same corresponding nonzero eigenvector.

Which of the following options is/are correct?

(Note: $I$ is the $n \times n$ identity matrix.)

• A. Determinant $(A-2 I)=0$
• B. Determinant $(B-2 I)=0$
• C. Determinant $(A+B-2 I)=0$
• D. Determinant $(A+B-4 I)=0$

Answer 1

We can use the identity: 

$$Av = \lambda v \quad \left \{  \lambda=2\right \}$$

$$(A-2I)v=0$$

Hence $v \in \text{Null}(A-2I) \implies |A-2I| =0$ Similarly for option (B) as well

$$(A+B)v=2v+2v$$

$$\therefore(A+B-4I)v =0$$

Hence $v \in \text{Null}(A+B-4I) \implies |A+B-4I| =0 \quad \quad\quad \left \{ |\cdot| \text{ denotes determinant}\right \}$ 

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Hence option{A,B,D} are accurate.

Answer 2

for option C and D

Ax = 2x -- eq1
Bx = 2x ---eq 2
adding 1 and 2
Ax+Bx = 2x + 2x 
(A+B)x = 4x
(A+B−4I)x=0

therefore
det(A+B-4I) =0