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# ISI MStat PSB 2009 Problem 1 | Nilpotent Matrices

This is a very simple sample problem from ISI MStat PSB 2009 Problem 1. It is based on basic properties of Nilpotent Matrices and Skew-symmetric Matrices. Try it !

## Problem- ISI MStat PSB 2009 Problem 1

(a) Let $$A$$ be an $$n \times n$$ matrix such that $$(I+A)^4=O$$ where $$I$$ denotes the identity matrix. Show that $$A$$ is non-singular.

(b) Give an example of a non-zero $$2 \times 2$$ real matrix $$A$$ such that $$\vec{x'}A \vec{x}=0$$ for all real vectors $$\vec{x}$$.

### Prerequisites

Nilpotent Matrix

Eigenvalues

Skew-symmetric Matrix

## Solution :

The first part of the problem is quite easy,

It is given that for a $$n \times n$$ matrix $$A$$, we have $$(I+A)^4=O$$, so, $$I+A$$ is a nilpotet matrix, right !

And we know that all the eigenvalues of a nilpotent matrix are $$0$$. Hence all the eigenvalues of $$I+A$$ are 0.

Now let $$\lambda_1, \lambda_2,......,\lambda_k$$ be the eigenvalues of the matrix $$A$$. So, the eigenvalues of the nilpotent matrix $$I+A$$ are of form $$1+\lambda_k$$ where, $$k=1,2.....,n$$. Now since, $$1+\lambda_k=0$$ which implies $$\lambda_k=-1$$, for $$k=1,2,...,n$$.

Since all the eigenvalues of $$A$$ are non-zero, infact $$|A|=(-1)^n$$. Hence our required propositon.

(b) Now this one is quite interesting,

If for any $$2\times 2$$ matrix, the Quadratic form of that matrix with respect to a vector $$\vec{x}=(x_1,x_2)^T$$ is of form,

$$a{x_1}^2+ bx_1x_2+cx_2x_1+d{x_2}^2$$ where $$a,b,c$$ and $$d$$ are the elements of the matrix. Now if we equate that with $$0$$, what condition should it impose on $$a, b, c$$ and $$d$$ !! I leave it as an exercise for you to complete it. Also Try to generalize it you will end up with a nice result.

## Food For Thought

Now, extending the first part of the question, $$A$$ is invertible right !! So, can you prove that we can always get two vectors from $$\mathbb{R}^n$$, say $$\vec{x}$$ and $$\vec{y}$$, such that the necessary and sufficient condition for the invertiblity of the matrix $$A+\vec{x}\vec{y'}$$ is " $$\vec{y'} A^{-1} \vec{x}$$ must be different from $$1$$" !!

This is a very important result for Statistics Students !! Keep thinking !!

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This is a very simple sample problem from ISI MStat PSB 2009 Problem 1. It is based on basic properties of Nilpotent Matrices and Skew-symmetric Matrices. Try it !

## Problem- ISI MStat PSB 2009 Problem 1

(a) Let $$A$$ be an $$n \times n$$ matrix such that $$(I+A)^4=O$$ where $$I$$ denotes the identity matrix. Show that $$A$$ is non-singular.

(b) Give an example of a non-zero $$2 \times 2$$ real matrix $$A$$ such that $$\vec{x'}A \vec{x}=0$$ for all real vectors $$\vec{x}$$.

### Prerequisites

Nilpotent Matrix

Eigenvalues

Skew-symmetric Matrix

## Solution :

The first part of the problem is quite easy,

It is given that for a $$n \times n$$ matrix $$A$$, we have $$(I+A)^4=O$$, so, $$I+A$$ is a nilpotet matrix, right !

And we know that all the eigenvalues of a nilpotent matrix are $$0$$. Hence all the eigenvalues of $$I+A$$ are 0.

Now let $$\lambda_1, \lambda_2,......,\lambda_k$$ be the eigenvalues of the matrix $$A$$. So, the eigenvalues of the nilpotent matrix $$I+A$$ are of form $$1+\lambda_k$$ where, $$k=1,2.....,n$$. Now since, $$1+\lambda_k=0$$ which implies $$\lambda_k=-1$$, for $$k=1,2,...,n$$.

Since all the eigenvalues of $$A$$ are non-zero, infact $$|A|=(-1)^n$$. Hence our required propositon.

(b) Now this one is quite interesting,

If for any $$2\times 2$$ matrix, the Quadratic form of that matrix with respect to a vector $$\vec{x}=(x_1,x_2)^T$$ is of form,

$$a{x_1}^2+ bx_1x_2+cx_2x_1+d{x_2}^2$$ where $$a,b,c$$ and $$d$$ are the elements of the matrix. Now if we equate that with $$0$$, what condition should it impose on $$a, b, c$$ and $$d$$ !! I leave it as an exercise for you to complete it. Also Try to generalize it you will end up with a nice result.

## Food For Thought

Now, extending the first part of the question, $$A$$ is invertible right !! So, can you prove that we can always get two vectors from $$\mathbb{R}^n$$, say $$\vec{x}$$ and $$\vec{y}$$, such that the necessary and sufficient condition for the invertiblity of the matrix $$A+\vec{x}\vec{y'}$$ is " $$\vec{y'} A^{-1} \vec{x}$$ must be different from $$1$$" !!

This is a very important result for Statistics Students !! Keep thinking !!

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