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QUESTION: 1

For a positive integer n, let denote the vector space of polynomials in one variable x with real coefficients and with degree __<__ n. Consider the map defined by T (p (x)) = p (x^{2}), then

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QUESTION: 2

Let A be a real 3 x 4 matrix of rank 2, then the rank of A^{t} A, where A^{t} deonles the transpose of A. is

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QUESTION: 3

Let V be the space of twice differentiable functions satisfying f" - 2f' + f = 0. Define by T(f') = (f'(0), f(0)), then T is

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QUESTION: 4

The determinant

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QUESTION: 5

Which of Ihe following matrices has the same row space as the matrix

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QUESTION: 6

The determinant of the n x n permutation

[x] denotes greatest integer function of x.

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QUESTION: 7

The row space of a 20 x 50 matrix A has dimention 13. What is the dimension of the space of solution of Ax = 0?

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QUESTION: 8

Given a permutation the matrix A is defined to be the one whose i^{th} column is the σ(i)^{th} column of the Identity matrix I. Which of the following is correct?

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QUESTION: 9

For the matrix A as given below, which of them satisfy A^{6} = I?

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QUESTION: 10

Let I denote the 4 x 4 Identity matrix. If the roots of the characteristic polynomial of a 4 x 4 matrix M

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QUESTION: 11

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QUESTION: 12

Let M be the set of all invertibel 5 x 5 matrices with entries 0 and 1. For each m ∈ M, let n_{1} (m) and n_{0}(m) denote the number of 1's and 0's in m respectively then

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QUESTION: 13

and b a non zero vector such that Mx = b for some Then the value of x^{T}b is

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QUESTION: 14

The matrix is a unitary matrix when α is

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QUESTION: 15

Let P be the vector space over all polynomials of degree less than 3 with real coefficients. Consider the linear transformation T : P → P defined by

T(a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3}) = a_{3} + a_{2}x + a_{1}x^{2} + a_{0}x^{3}

Then the matrix representation of M of T with respect to the ordered basis {1, x, x^{2},x^{3}} satisfies

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QUESTION: 16

The largest eigenvalue of the matrix

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QUESTION: 17

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QUESTION: 18

Let a,b,c,d be distinct non zero real numbers with a + b = c + d. Then an eigenvalue of the matrix

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QUESTION: 19

The system of linear equations

x - y + 2z = b_{1}

x + 2y - z = b_{2}

2y - 2z = b_{3}

is inconsistent when (b_{1}, b_{2}, b_{3}) equals

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*Multiple options can be correct

QUESTION: 20

Let X and Y are n x n matrices with real entries, then which of the following is(are) true?

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