Linear Algebra - Study Mode
[#71] For a matrix [left[ { ext{M}}
ight] = left[ {x08egin{array}{*{20}{c}}
{frac{3}{5}}&{frac{4}{5}} \
{ ext{x}}&{frac{3}{5}}
end{array}}
ight],] xa0 xa0the transpose of the matrix is equal to the inverse of the matrix, [M] T = [M] -1 . The value of x is given by
Correct Answer
(A) [ - frac{4}{5}]
[#72] Consider a matrix P whose only eigenvectors are the multiples of [left[ {x08egin{array}{*{20}{c}}
1 \
4
end{array}}
ight].] Consider the following statements: I. P does not have an inverse. II. P has a repeated eigen value. III. P cannot be diagonalized. Which one of the following options is correct?
Correct Answer
(D) Only II and III are necessarily true
[#73] Let [{ ext{P}} = left[ {x08egin{array}{*{20}{c}}
3&1 \
1&3
end{array}}
ight].] xa0 Consider the set S of all vectors [left( {x08egin{array}{*{20}{c}}
{ ext{x}} \
{ ext{y}}
end{array}}
ight)] such that a 2 + b 2 = 1 where [left( {x08egin{array}{*{20}{c}}
{ ext{a}} \
{ ext{b}}
end{array}}
ight) = { ext{P}}left( {x08egin{array}{*{20}{c}}
{ ext{x}} \
{ ext{y}}
end{array}}
ight).] xa0 Then S is
Correct Answer
(D) an ellipse with minor axis along [left( {x08egin{array}{*{20}{c}}
1 \
1
end{array}}
ight)]
[#74] In the given matrix [left[ {x08egin{array}{*{20}{c}}
1&{ - 1}&2 \
0&1&0 \
1&2&1
end{array}}
ight],] xa0 one of the eigen values is 1. The eigen vectors corresponding to the eigen value 1 are
Correct Answer
(B) {α(-4, 2, 1) |α ≠ 0, α [ in ] R}
[#75] Choose the CORRECT set of functions, which are linearly dependent.
Correct Answer
(C) cos 2x, sin 2 x and cos 2 x