GATE CSE 2018 | Question: 26

Question

Consider a matrix P whose only eigenvectors are the multiples of $\begin{bmatrix} 1 \\ 4 \end{bmatrix}$.

Consider the following statements.

1. P does not have an inverse
2. P has a repeated eigenvalue
3. P cannot be diagonalized

Which one of the following options is correct?

• A. Only I and III are necessarily true
• B. Only II is necessarily true
• C. Only I and II are necessarily true
• D. Only II and III are necessarily true

Comments

1. A necessary and sufficient condition for a matrix $A_{n\times n}$ to be diagonalisable is that the matrix must have $n$ linearly independent eigen vectors.
2. A sufficient condition for a matrix $A_{n\times n}$ to be diagonalisable is that the matrix must have $n$ linearly independent eigen values.

 

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@kushagra     what is meaning of linearly independent eigen values?

And why 2nd statement is only sufficient condition?

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@Gyanu I am not very sure of it. Maybe while I was going through vectors I misunderstood something and wrote those points. I felt that $2)$ one is not right. I Strike that for now but if got something I will update it.

Answer 1

so that option D is correct.

Answer 2

I think this is the correct solution, here goes:

First of all, the eigen vector is a 2 x 1 matrix, suggesting that there are 2 eigen values. They're either distinct, or not.
If they're distinct, then the eigen vectors corresponding to them would be linearly independent, hence P would be diagonalizable.

But in the question, it is mentioned that the only eigen vectors it has are multiples of [1 4]. No pair of vectors from this set would be linearly dependent, hence P cannot be diagonalized, and the two eigen values are the same.

Comments

I didn't get this.

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This might help

If eigen values are distinct then eigen vectors linearly independent and matrix can be diagonalized but if eigen values are repetitive and eigen vectors linearly dependent in such case matrix can not be diagonalized but if eigen vectors are independent for repeatitive and distinct eigen value  then matrix can be diagonalized.

 

Answer 3

Eigenvectors of matrix P =  k$\begin{bmatrix}1 \\ 4 \end{bmatrix}$ 

Here,

• k denotes free variable
• $\begin{bmatrix}1 \\ 4 \end{bmatrix}$ denotes matrix P has only one Linearly independent eigenvector

Statement I. For matrix to have inverse, we need to know either λ=0 or not

Statement II. Since, eigenvector is multiple of k , so it must have repeated eigen values

Statement III. A matrix can be diagonalizable iff it has n LI eigen vectors. 

So only statement II, III are true

Comments

why we can’t diagonalize a matrix if we don’t have $n$ LI eigen vector ?