GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 4

Answer 1

λ^2 (λ-3) (λ+2)^3 (λ-4)^3
         From the above equation once can
         obtain the dimension that will be
         Sum of powers of the above equation= 2+1+3+3 = 9
            λ=0,3,-2,4
            Now, λ=0
                                  (A - λ)X = 0 ;
                                   AX =0 ;
--> Atmost 2 Linear independent eigen vector corresponding to λ=0
--> Atleast 1 Linear Independent eigen vector corresponding to λ=0
Rank = Number of column – Number of free variable (Nullity )
        Atmost 2 :
Rank could be = 9 – 2 = 7
       For Atleast 1 :
  Rank could be = 9 – 1 = 8
  So the possible could be ( 7 , 8 ) .

Option B and C correct one .

Comments

Pardon me for assuming lambda as “k”

Now , Why I am taking for only k = 0 , to consider for the rank why not 4 , or any non zero value ?

As we know :

[A – kI][x] = 0 ; where x reprsent the Eigen Vector Combination corresponding to k .

Now if we have k =0 ;

Ax = 0 → this seems to be familiar with the system of linear equation of homogenous solution . Now if we have x as the combination of Linear Independent Eigen Vector this is fairly equals to say that there will be the free varibles or Linear dependent columns in a or nullity of A .

If we get the Nullity then we can easily get the Rank by Rank -Nullity Theorem .

That’s why for k =0 , we can conclude about the rank fairly easily because it does not change the A in [A-kI] bcz k = 0.

And for  k^2 :

Geometric muiltiplicity <= Algebraic Muiltiplicity .

that mean for AM = 2 , you can have ( 1 , 2 ) as the GM .

Now why we won’t tell about with other non zero  eigen value ?

For k  =  x ( x is any non zero value )

[A – kI][x] = 0 ; ----------- 1

Ax = 0 ------------------- 2

comparing only A and A-k ;

A != A-k ( They will not be equal so A-k would be diffrent matrix )

So we can’t conclude about rank of A from ( k – x ) .

Hence Proved that we can only infer  rank of A from the eigen value which is equal to 0 .

Note : If their is No Eigen value = 0 , then this mean that the rank will be full .

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Thanks I understood it Now.

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Let K be any Eigen value.

[A- kI] [x] = 0

Rank and nullity can be obtained of matrix A- kI

if k = 0 , then rank and nullity getting of A

 

Instead of keeping in mind at k= 0 we could only get rank of A. We can see how it's actually happening.

Answer 2

Here the eigen values are = 0,0,3,-2,-2,-2,4,4,4 

Sum of powers = total number of column = 9 , hence n=9.

Here they have asked for rank of A , so we will use eigen value (λ) =0 for this because for this eigen value A and (A- λI) is same. Now for this eigen value we can have 1 or 2 linearly independent eigen vectors or solution , hence there can be either 1 or 2 free column.

using rank nullity theorem , Rank+nullity= Total number of column.

so Rank can be = 9-1 , 9-2 .

ANSWER=8,7.