BSc Sem III 2023 2024 2024 MATHS II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q2 For Q.1, Q.2 and Q. 3 attempt any one subquestion (each 8 marks) from part (a), and any two subquestions (each 6marks) from part (b)
2. Q3 For Q.4, attempt any three. (each 5 marks) any one. [each 1)Let A, B be two row equivalent matrices then prove that A is invertible iff B is invertible 2)Let A, B be the matrices of order mxn then prove that A, B are row equivalent off there existsan invertible matrix P such that B=PA
   • (b) Attempt any two. [each 6Mks] 1)Check whether the following system of equation is consistent and if so, find the solution set
3. Q2 Check whether the following are elementary matrices 3)Express the following matrices and their inverses as product of elementary matrices ry
   • (a)Attempt any one. [each 8Mks] 1)Let V be a real vector space and W be a subspace of V then prove that
   • i) v+W=W iff v belongs to W ii) iff vi-v2 belongs to W
4. Q2 Linearly independent set and Prove that Superset of linearly dependent set is linearly
   • (b) Attempt any two. [each 6Mks] 1)Check whether the given vector v belong to L(S) linear span of S in the following space
5. Q2 Check whether = 1,1), (3,2), (4,6) }in R? are Linearly independent or not? 3)Prove that (R,+,*)is a real vector space with respect to usual addition+ and multiplication
6. Q3 (a) Attempt any one. 8Mks]
7. Q1 Let A be A nxn real matrix.If det(A) 40 then prove that A is invertible and A~? = any two. [each 6 Mks] basis of row space and column space 2)Prove that if A is a square matrix then det(A) = det (A*) ‘ 3)Find using adjoint of A
8. Q4 Attempt any three. [each 5 Mks] 1)Show that the following system of equations have infinitely many solutions applying Gauss 2)Prove that the following system of equations have no solutions
9. Q3 Check whether Q the set of rational number are vector space over R the set of real numbers with respect to usual addition and multiplication
10. Q4 Prove that S = {(1,0), a basis of R? 5)Find the determinant of } 4 6 —5| using Laplace expansion along colmn 3 6)Solve AX = B using LU decomposition method where 1

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