GATE CSE 2007 | Question: 25

Question

Let A be a $4 \times 4$ matrix with eigen values -5,-2,1,4. Which of the following is an eigen value of  the matrix$\begin{bmatrix} A & I \\ I & A \end{bmatrix}$, where $I$ is the $4 \times 4$ identity matrix?

• A. $-5$
• B. $-7$
• C. $2$
• D. $1$

Comments

Answer here!   

---

Ans

---

Thanks @goku4199 Besst Explanation

Answer 1

Ans will be C.

Answer 2

applying a trick on the matrix:

if you simply add the R1 elements.......you get A+I
then adding R2 elements....you get A+I
we are getting same ans i.e. A+I in both cases......so we can conclude that....one of the eigen values is A+I for sure
[this trick can be applied to other matrices too]
now, taking into account the given eigen values for A......evaluate A+I...i.e. add 1 to each value.....
you get -4,-1,2,5.............thus we get option C] 2 as ans