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 !!

Similar Problems and Solutions

ISI MStat PSB 2008 Problem 10 — Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Related

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 !!

Similar Problems and Solutions

ISI MStat PSB 2008 Problem 10 — Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Related

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