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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.
Find the general solutions of the systems whose augmented matrices is given?

[ 3 -4 2 0 ]
[ -9 12 -6 0 ]
[ -6 8 -4 0 ]
Find the sum of squares of all numbers x such that both expressions x^2+5x and x+(1/x) are integer numbers.

Let T : R3 —> R3 be defined by

T(xi, x2, x3) = (3x1 + x3, - 2x1 + x2,

- x1 + 2x2 + 4x3)

Show that T^(-1) exists. Give the expression for T^(-1)(x1 , x2, x3) for T above.

There is no co-ordinate transformation that transforms the quadratic form x² + y²+ z² to xz + yz. True or false

Are there values of (a) belongs to C for which the matrix [
1 0 0
0 -1/√2 1/√2
0 1/√2. a
is unitary? Justify your answer
Find the range space and a basis for the
kernel of the linear transformation
T : R4 ->R4 defined by
T(x1, x2, x3, x4) = (x1 - x2, x2 - x3, x3 - x4, x4 - x1).
Reduce the quadratic form Q=x1²+2x2x3 to canonical form and hence find its nature, rank, index and signature.
Which of the following statements are true and
which are false ? Give reasons for your
(a) Any square matrix with real entries is either symmetric or skew-symmetric or a linear combination of such matrices.
(b) If a linear operator has an eigenvalue 0,
then it cannot be one-one.
(c) If f : V --> K is a non-zero linear functional
and V a vector space of dimension n, then
there are n - 1 linearly independent vectors
v belongs to V such that f(v) = 0.
(d) Every binary operation on Rn is
commutative, for all n belongs to N.
(e) IAdj (A)I = IAI for all A belongs Mn(R).
Consider the real vector space
A = {(a, b, c, d) I a, b, c, d belongs to R,2a + 3b = c + d}.
Find dim A. Also find two distinct subspaces
B1. and B2 of R⁴ such that
A (direct sum) B1=R⁴=A(direct sum)B2
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