Question

Eigenvalue of matrix $A$ ,  $\begin{bmatrix} 2 &7 &10 \\ 5& 2 & 25\\ 1& 6 &5 \end{bmatrix}$    is   $-9.33$                                    

other eigenvalue is

1) $18.33$

2) $-18.33$

3) $18.33-9.33 i$

4) $18.33+9.33i$

Answer 1

Ans will be (A)

trace of matrix = sum of eigen values

say here are 3 eigen values are -9.33, X,Y

2+2+5 = -9.33+X+Y

or, X+Y= 18.33....................(i)

Now, determinant of matrix= product of eigen values

determinant of matrix =2(10-150) - 7(25-25) + 10(30-2) =0

product of eigen values = -9.33 .X.Y

or, -9.33.X.Y=0

or, X.Y=0

So, we can say one of the eigen value is 0, say, X=0

then from equation (i) we can say Y=18.33

Answer 2

Let x and y be other two eigen value 
sum of eigen value = sum of diagonals 
x + y +(-9.33)=2+2+5 
x+y=18.33
x=18.33-y

product of eigen value = determinant of matrix
xyz = 0
(18.33-y)y(-9.33) =0 
so y=0 or 18.33

1)18.33 is the answer.
 

Answer 3

Rank of matrix is 2 since column1 and column3 are dependent. So one of the eigen values is 0.

Sum of eigen values= trace of matrix=2+2+5=9

-9.33+0+a=9 ==> a=18.33

option A)