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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 !

(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 \(\vec{x}\).

Nilpotent Matrix

Eigenvalues

Skew-symmetric Matrix

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.

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 !!

What to do to shape your Career in Mathematics after 12th?

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

- What are some of the best colleges for Mathematics that you can aim to apply for after high school?
- How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
- What are the best universities for MS, MMath, and Ph.D. Programs in India?
- What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
- How can you pursue a Ph.D. in Mathematics outside India?
- What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

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