Gate 2016 CE Set 1

Question

If the entries in each column of a square matrix M add up to 1, then an eigen value of M is

A) 4  B) 3  C) 2  D) 1

Comments

Take identity matrix as an example where the condition " the entries in each column of a square matrix M add up to 1 " holds true. In such case eigen values are 1.

D)

---

Am getting 1 by taking examples.

Smallest example is order 1 matrix

$\begin {bmatrix} 1 \end{bmatrix}$

Eigen value is 1.

Answer 1

Consider the 2×2 matrix A=$\begin{bmatrix} a &b \\ c& d \end{bmatrix}$

$\lambda^2 -trace(A)  \lambda+det(A)=0$

$\lambda^2 -(a+d  )\lambda+(ad-bc)=0$

Put $\lambda$=1

$1-a-d+ad-(1-d)(1-a)=0$

$\text{as we know that a+c=1 and b+d=1}$

$1-a-d+ad-1+d+a-ad =0$

$0=0$ other option $\lambda=2,3,4$doesn't satisfy

$\mathbf{(D )ans}$

Comments

i think after step 2 if u dont put the value of ƛ=1... u can enter into an equation where u can take (ƛ-1) common=0 which will give ƛ=1... ultimately answer.

Answer 2

 

Answer is 1 i.e. Option d . Here's the explanation :-