BSc Mathematics SEM III 2022 2023 Oct 2023 MATHEMATICS II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 (a) Attempt any one. [each 8Mks]
   • 1)Prove that row equivalence relation on the set of matrices
   • 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
   • 2) Check whether the following matrices are elementary
   • 3) Express the following matrices and their inverses as a product of elementary
2. Q2 Attempt any one. [each 8Mks] 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) v+W=v'+W iff v-v' belongs to W
   • 2)Define Linearly independent set and Prove that Subset of linearly independent set is linearly
   • (b)Attempt any two. 6Mks}
   • 1) Check whether the vector is belong to L(S) for v=(1, 3,2), S={(2,1,
   • 1),(3,2,0),(4,0,3)} in R?
   • 2) Prove that the set of all points on the plane passing through origin form a subspace in R? with respect to usual addition and scalar multiplication ; 3) +,.) Be a vector Space and W, be two subspace of V then prove that is also a subspace of
3. Q3 Attempt any one. [each
   • 1) Let A be A nxn real matrix.If det(A) #0 then A is invertible and A”! =(1/det(A) )x
   • 2) Prove that det(V,) =[]
   • (b) Attempt any two. [each 6 Mks]
   • 1) Prove that determinant of upper or lower trianguiar matrix is Product of diagonal entries
   • 2) Find the adjoint of the following 7
   • 3) Find row rank, column rank and rank of the following matrix using row echelon form also find basis of row Space and column space. 2
   • 1) Show that the following system of equations have infinitely many solutions applying
   • 2)Express the following matrix in row echelon form in two different way. 31
   • 3) Prove that (R?,+,.) Is a vector Space with respect to addition and scalar multiplication in R? for + and. As follows (a, b) +(c, d) and b) ab) for aeR and (a, b),(c,
   • d)eR?
   • 4) Find the basis and dimension of X1-2x2+3x3=0} of
   • 5) Find the determinant of following matrix. -9 73
   • 6) Find the nulity of following coefficient matrix for given system of equation

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