Linear Algebra - Study Mode
[#131] The linear operation L(x) is defined by the cross product L(x) = b × X, where b = [0 1 0] T and X = [x 1 x 2 x 3 ] T are three dimensional vectors. The 3 × 3 matrix M of this operation satisfies [{ ext{L}}left( { ext{x}}
ight) = { ext{M}}left[ {x08egin{array}{*{20}{c}}
{{{ ext{x}}_1}} \
{{{ ext{x}}_2}} \
{{{ ext{x}}_3}}
end{array}}
ight].] Then the eigen values of M are
Correct Answer
(D) i, -i, 0
[#132] What are the value of k for which the system of equations: (3k - 8)x + 3y + 3z = 0 3x + (3k - 8)y + 3z = 0 3x + 3y + (3k - 8)z = 0 has a not-trivial solution?
Correct Answer
(D) [{ ext{k}} = frac{2}{3},,frac{{11}}{3},,frac{{11}}{3}]
[#133] The rank of the following matrix is [left( {x08egin{array}{*{20}{c}}
1&1&0&{ - 2} \
2&0&2&2 \
4&1&3&1
end{array}}
ight)]
Correct Answer
(B) 2
[#134] If [{ ext{A}} = left[ {x08egin{array}{*{20}{c}}
{2 + { ext{i}}}&3&{ - 1 + 3{ ext{i}}} \
{ - 5}&{ ext{i}}&{4 - 2{ ext{i}}}
end{array}}
ight],] xa0 xa0 then AA ∗ will be (where, A ∗ is the conjugate transpose of A)
Correct Answer
(C) Hermitian matrix
[#135] Consider the following system of linear equations: 3x + 2ky = -2 kx + 6y = 2 Here, x and y are the unknown and k is a real constant. The value of k for which there are infinite number of solutions is
Correct Answer
(C) -3