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 !!


ISI MStat PSB 2008 Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

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