Linear Algebra Answers

Find the minimal polynomial of the linear
operator T : R4 -->R4, defined by
T(a, b, c, d) = (a, a + b, c, 2d).

Which of the following statements are True and
which are False ? Justify your answer with a
short proof or by a counter-example.

(a) The operation *, defined by x * y = log (xy) is
a binary operation on S, where
S={xER x>0}.

(b) If a and b are eigenvalues of two n x n
matrices A and B respectively, then a + b is
an eigenvalue of A + B.

(c) If S and T are linear transformations such
that SoT is defined and is 1 — 1, then S is
1 — 1.

(d) T : R³- R³: T((x1, x2, x3), (y 1 , y2, y3)) =
(x1 + x2 + x3) . (y1 + y2 + y3)
is an inner product on R³.

(e) {India, — 5, Jamila} is a set.

If M is a singular Matrix, is there a value of k∈N for which kM will be non singular? Give reasons for your answer

Let S={(1,4),(0,3)} be a subset of R^2(R).prove that (2,3) belongs to L(S)

Find a basis for the null space, row space and column space of matrix

if 2,-1,-3 are the eigen values of the matrix A then find the eigen values of the matrix A Inverse

Find the sum and product of the matrix of the given eigen values of the matrix
A=2 1 2
1 3 1
2 2 -6

Solve the system of following equation by guass elimination method
3x1+2x2+6x3=2
6x1-7x2-11x3=8
-5x1+9x2+3x3=10

Choose h and k such that the system below has (a) no solution, (b) a unique solution, and (c)
many solutions.

x1 + 3x2 = 2

3x1+ hx2 = k.