Optionals IAS Mains Mathematics: Questions 93 - 98 of 283

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Question 93

Hermitian, Skew-Hermitian
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Appeared in Year: 2010

Describe in Detail

Essay▾

Find a Hermitian and a skew-Hermitian matrix each whose sum is the matrix.

(CS main Paper 1)

Explanation

Let

Which is Hermitian Matrix.

Again

Which is skew-Hermitian matrix.

Now,

Matrix has been expressed as the sum of Hermitian and skew-Hermitian matrix.

Question 94

Linear Transformations
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Appeared in Year: 2007

Describe in Detail

Essay▾

Let be the vector space of all polynomial with real coefficient of degree less than or equal to two considered over the real field , such that and . Determine a basis for and hence its dimension. (CS main Paper 1)

Explanation

Let

be the given vector space of all polynomials with real coefficient of degree less than or equal to two considered over the real field , such that and .

To determine a basis for .

Given that

And

the given vector space becomes

To find a basis for a finite subset of

s. t

Let,

Then where .

From eq. , we have .

i. e. spans .

Let s. t

is L. I subse

… (9 more words) …

Question 95

Eigenvalues and Eigenvectors
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Appeared in Year: 2013

Describe in Detail

Essay▾

Let where is a cube root of unity. If denote the eigen values of show that| . (CS main Paper 1)

Explanation

Given

is cube root of unity.

and and .

Eigenvalue of are

Now

Question 96

Eigenvalues and Eigenvectors
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Appeared in Year: 2011

Describe in Detail

Essay▾

Let be the eigen value of a square matrix with corresponding eigen vector . if is Matrix Similar to Show that the eigen value of are same as that of and eigen vectors of A. (CS main Paper 1)

Explanation

is Matrix Similar to (Given)

So, inverible Matrix such that

Matrices and have the same characteristics polynomial and hence the same set of given values.

Now, Let is given value of Matrix

Then

with

is eigen vector of corresponding to eigen value . This is relation b/w eigenvector of two similar matrices.

Question 97

Matrix of a Linear Transformation
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Appeared in Year: 2013

Describe in Detail

Essay▾

Let denote the vector space of all polynomials of degree at most and be a linear transformation given by

Find the matrix of with respect to the bases and of respectively. Also, find the null space of T. (CS main Paper 1)

Explanation

Given be a linear transformation defined by

And basis of respectively.

Now,

If is matrix of w. r. t. Basis of .

Now to find Null Space of T

Let (Null Space of )

Question 98

Cayley-Hamilton Theorem
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Appeared in Year: 2014

Describe in Detail

Essay▾

Verify Cayley-Hamilton Theorem for the Matrix and hence find its inverse.

Also find the Matrix represented by (CS main Paper 1)

Explanation

Given

The characteristic equation of is

We have to prove that satisfies the equation

Now,

=

=

So, Cayley – Hamilton Theorem is Satisfied.

Now,

It is Inverse of Matrix

Now, We find the Matrix represented by

We have and

=

=

So, =

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