Name of the University : North Maharashtra University
Degree : B.Sc
Department : Mathematics
Name Of The Subject : Algebra And Trigonometry
Document type : Question Bank
Website : nmu.ac.in
Download Model/Sample Question Papers : https://www.pdfquestion.in/uploads/nmu.ac.in/5294-F.Y.%20B.%20Sc.%20(Mathematics)%20Question%20Bank-I.pdf
NMU Algebra And Trigonometry Model Paper
Unit – 01
Part – I
Adjoint and Inverse of Matrix, Rank of a Matrix and Eigen Values and Eigen Vectors
1) find minor and cofactor of a11, a23 and a32
2) If A = – find adj A
3) If A = –
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4) If A = – and B = – find – (AB)
5) If A = find – (A)
6) Find rank of A =
7) Find the characteristic equation and eigen values of A = –
8) Define characteristic equation of a matrix A and state Cayley-Hamilton
9) Define adjoint of a matrix A and give the formula for A-1 if it exist.
10) Define inverse of a matrix and state the necessary and sufficient condition for
11) Compute E12(3) , E’ 2(3) of order 3
12) If A = – and B = – find (AB)-1
13) If A = – then – (A) is — a) 0 b) 1 c) 2 d) 4
14) If A = – then which of the following is true –
a) adjA is nonsingular b) adjA has a zero row
c) adjA is symmetric d) adjA is not symmetric
15) If A = – then which of the following is true –
a) A2 = A b) A2 is identity matrix
c) A2 is non-singular d) A2 is singular
16) If A = – and B = – Statement I : AB singular
17) If A is a square matrix, then A-1 exists iff
a) A > 0 b) A < 0 c) A = 0 d) A – 0
18) If A = – then A(adj A) is
19) If A is a square matrix of order n then KA is a) K A b) ( )n
20) Let I be identity matrix of order n then
a) adj A = I b) adj A = 0 c) adj A = n I d) None of these
21) Let A be a matrix of order m x n then A exists iff
a) m > n b) m < n c) m = n d) m – n
22) If AB = – and A = – then det. B is equal to
23) If A = – and A-1 = – then x = — a) -1/2 b) -1/2 c) 1 d) 2
24) If A = – and n- N then An is —-a)
25) If A = – then adjA is a) 10 b) 1000 c) 100 d) 110
26) If a square matrix A of order n has inverses B and C then
a) B- C b) B = Cn c) B = C d) None of these
27) If A is symmetric matrix then
a) adjA is non-singular matrix b) adjA is symmetric matrix
c) adjA does not exist d) None of these
28. If A = 0 and B, C are matrices such that AB = AC then
a) B = C
b) B = A
c) B = C
d) C = A
29. If matrix A is equivalent to matrix B then
a) p(A) = p(B)
b) p(A) > p(B)
c) p(A) = p(B)
d) None of these
30. If A is a matrix of order m x n then
a) p(A) =min{m,n}
b) p(A) = min{m,n}
c) p(A) = max{m,n}
d) None of these
31. If A is a matrix and y is some scalar such that A – ?I is singular then
a) y is eigen value of A
b) y is not an eigen value of A
c) y = 0
d) None of these
Part – II
4 Marks Questions :
1. For a non-singular square matrix A of order n , prove that adj (adj A) = A(n -1)2
2. For a non-singular square matrix A of order n , prove that adj {adj (adjA)} = 3 3 2A n – n + A-1
3. Compute the elementary matrix [E2(-3)]-1. E31 (2) . E ‘ 21 (1/2) of order 3
4. Compute the matrix E ‘ 2 (1/3) . E31 . [E2(-4)]-1 for E-matrices of order 3
5. If y is a non-zero eigen value of a non-singular matrix A, show that 1/y is an eigen value of A-1
6. If y= 0 is an eigen value of a non-singular matrix A, show that A /y is an eigen value of adj A.
7. Let k be a non-zero scalar and A be a non-zero square matrix, show that if y is an eigen value of A then ?k is an eigen value of kA.
8. Let A be a square matrix. Show that 0 is an eigen value of A iff A is singular.
9. If A, B are matrices such that product AB is defined then prove that(AB)’ = B’A’
10. If A = [ aij ] is a square matrix of order n then show tha A(adjA) = (adjA)A = A I
11. Show that a square matrix A is invertible if and only if A = 0
12. If A, B are non-singular matrices of order n then prove that AB is non-singular and (AB)-1 = B-1 A-1
13. If A, B are non-singular matrices of same order then prove that adj(AB) = ( adjB ) ( adjA )
14. If A is a non-singular matrix then prove that (An)-1 = (A-1)n , ?n?N