Find the inverse of the matrix B in part a) by using Cayley-Hamilton theorem.

For the following matrices, check whether there exists an invertible matrix P such that P^-1AP is diagonal. When such a P exists, find P.

Find the solutions to the following system of equations by reducing the corresponding augmented matrix to row-reduced echelon form. 

2a+3b+4c+d=8

a+2b+2c+2d=3

a-b+c+3d = 3

Show that, if A is any n x n matrix with real entries, then there is a n x n symmetric matrix S and a n x n skew symmetric matrix S' such that A = S + S'. 

Is the matrix of the linear operator T non-singular? Justify your answer.

Write down the matrix of T on W w.r.t the basis S.

Check that T(W)⊂ W.

Let W = [S] and let T: V → V be the function defined by

Check that T is a linear transformation on V.

Check that S is a linearly independent set over R. (Hint: Consider the equation

Let V be the set of all functions that are twice differentiable in R and

S = {cosx, sinx, xcosx,xsinx}.

Find the direction cosines of the perpendicular from the origin to the plane 

Check that (fog)(x) = x for x ∈ R \{2} and (gof) (x) = x for x ∈ R \{-1

Check that g: R\{2} → R given by g(x) = is well defined and 1 - 1. Further, check that g(x) -1 for any x ∈ R.

Check that f(x) ≠ 2 for any x ∈ R.

Check that f(x) is well defined and 1 - 1.

Consider the funtion f: R\{-1} → R defined by f'(x) = . 

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

i) The function  defined by f(x) = cosx is 1-1.

ii) The operation * defined by x*y = log(xy) is a binary operation on S, where S is the set {x ∈ R x > 0}.

(iii)The set {(x₁.x₂,...,xn) X₁,X₂,...,xn ∈ R,X₁ = 2x2+3} is a subspace of Rⁿ

iv) There is no 7 x 5 matrix of rank 6.

v) If V and V' are vector spaces and T: V→ V' is a linear transformation, then whenever u₁,u₂,..., u_k are linearly independent, T_u₁, T_u2, ... Tu_k are also linearly independent.

vi) If V is a vector space and T: V → V is a linear operator with det(T) = 0, then T is not diagonalisable.

vii) The degree of the minimal polynomial of a 3 x 3 matrix is at most 2.

viii) For any 2 x 2 matrix A, Adj (A') = (Adj(A))'.

ix) The only matrix which is both symmetric and skew-symmetric is the zero matrix.

x) There is no co-ordinate transformation that transforms the quadratic form x²+y² +z² to the quadratic form xz+yz.

बिरजे-वीटा विधि के प्रयोग से अंतराल [2,3] में समीकरण :

x²-2x-5=0

का एक मूल ज्ञात कीजिए। केवल एक ही पुनरावृत्ति दीजिए।

10-^5, -1≤ x ≤ 1 के मैक्लॉरिन प्रसार में कितने पद nहोने चाहिए ताकि त्रुटि e^x से कम रहे?

h = 0.1 लेकर तृतीय कोटि टेलर श्रेणी विधि से x = 0.1 पर आदिमान समस्या :

y'=x-y, y(0) = 1

का हल ज्ञात कीजिए।