TIFR CSE 2018 | Part A | Question: 14

Question

Let $A$ be an $n\times n$ invertible matrix with real entries whose row sums are all equal to $c$. Consider the following statements:

1. Every row in the matrix $2A$ sums to $2c$.
2. Every row in the matrix $A^{2}$ sums to $c^{2}$.
3. Every row in the matrix $A^{-1}$ sums to $c^{-1}$.

Which of the following is TRUE?

• A. none of the statements $(1), (2), (3)$ is correct
• B. statement $(1)$ is correct but not necessarily statements $(2)$ or $(3)$
• C. statement $(2)$ is correct but not necessarily statements $(1)$ or $(3)$
• D. statement $(1)$  and $(2)$ are correct but not necessarily statement $(3)$
• E. all the three statements $(1), (2),$ and $(3)$ are correct

Comments

yes..

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All the prosperity are satisfied but by taking example, any mathematical way to do it ?

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i solved it using abcd as values and applying condition ad-bc !=0 and a+b=c+d='c'

Answer 1

 

we have matrix A, whose rows sums are all equal to C.

i,e; C is one of the eigen value of matrix A (eigen value property)

 

By properties of eigen values,  we know

1. 2C is eigen value of matrix 2A

2. $C^{2}$  is eigen value of matrix $A^{2}$

3. $C^{-1}$ is eigen value of matrix $A^{-1}$

 

All three properties are satisfied

so, option (E) is the correct answer.