Indian Institute of Technology Ropar Department of Mathematics
MA 102-Linear Algebra and Integral Transforms Second Semester of Academic Year 2018-2019
Tutorial Sheet - 5
1. Find the row reduced echelon form of the following matrix:
M =
1 2 −3 1 2 2 4 −4 6 10 3 6 −6 9 13
.
2. Find the dimension of the row space and the column space of the following matrix:
M =
1 −3 4 −2 5 4
2 −6 9 −1 8 2
2 −6 9 −1 9 7
−1 3 −4 2 −5 −4
Are they equal?
3. For the following matrices A:
1 4 5 2 2 1 3 0
−1 3 2 2
1 4 5 6 9
3 −2 1 4 −1
−1 0 −1 −2 −1
2 3 5 7 8
(a) Find a basis for the null space.
(b) Find a basis for the row space by reducing the matrix to row reduced echelon form.
(c) Find a basis for the column space.
(d) Find a basis for the row space of A consisting entirely of row vectors of A.
4. Consider the linear transformation T on R2 defined by T(x, y) = (2x−3y, x+ 4y) and the basisE ={(1,0),(0,1)} and S={(1,3),(2,5)}.
(a) Find the matrix A representing T relative to the bases E and S.
(b) Find the matrix B representing T relative to basis S and E.
(c) How the matrixA and B related.
5. Let V = P(t), the vector space of polynomials over K = R or C. Let F be the mapping fromV to V defined by
F(a0+a1t+a2t2+· · ·+antn) = a0t+a1t2+· · ·+antn+1 (a) Show that F is a linear mapping and non-singular.
(b) Is F invertible?
6. Let G:R2 →R3 be defined byG(x, y) = (x+y, x−2y,3x+y).
(a) Show that G is non-singular.
(b) Find a formula for G−1.
7. If A and B are square matrices of order n and λ is any scalar, then show that (i) tr(λA) = λ tr(A).
(ii) tr(A+B) = tr(A)+ tr(B).
(iii) tr(AB)= tr(BA).
8. If A is a square matrix such that A2 =A, the A is called idempotent.
if A and B is idempotent matrices and commute, then Prove that (a) AB is idempotent.
(b) If AB=BA= 0, then A+B is idempotent.
(c) If A+B =I, (assume that B idempotent is not given) then show B is idem- potent and AB=BA= 0.
9. Show that x = −9 is a root of
x 3 7 2 x 2 7 6 x
= 0. Find the other two roots of the equation.
10. Obtain the rank of the following matrices:
M =
−1 2 0 3 7 1 5 9 3
and
N =
1 1 2 3
1 3 0 3
1 −2 −3 −3
1 1 2 3
11. Show that the rank of the transpose of a matrix is equal to the rank of the original matrix, i.e, r(A) = r(At).
12. Show the followings:
(i) Eij−1 =Eij. (ii)
Ei(k)−1
= Ei(1/k), where k 6= 0.
(iii)
Eij(k)−1
= Eij(−k), where k 6= 0.
(iv) Eij0 −1
=Eij0 . (v)
Ei0(k)−1
= Ei0(1/k), where k 6= 0.
(ii)
Eij0 (k)−1
= Eij0 (−k), where k 6= 0.
13. Show that every non-singular matrix is a product of elementary matrices.
14. Show that if A∼B, then r(A) = r(B).
15. Compute the following matrix E23(−1)·E31·E240 ·E3(2) for an elementary matrix of order 4.
16. Find the rank of A+B, AB, BAwhere A=
1 1 −1
2 −3 4 3 −2 3
; B =
−1 −2 −1
6 12 6
−5 10 5
.
17. Reduce the matrix to normal form
(i)
1 1 2 3
1 3 0 3
1 −2 −3 −3
1 1 2 3
; (ii)
1 −1 3 6
1 3 −3 −4
5 3 3 11
.
18. Find the inverse of A=
1 −1 2 −3
4 1 0 2
0 3 0 4
0 1 0 2
, using elementary operations.
19. Obtain a non-singular matrix X such that XA=I, whereA=
1 2 3
2 4 5
3 −5 6
.
20. Reduce the matrix A =
1 2 0 4 3 −5 1 1 5
to I3 by a finite sequence of elementary row operations and hence expressA as a product of elementary matrices. Find A−1
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