Engineering Maths · all questions

Linear Algebra
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Q121

The eigen values of a symmetric matrix are all

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Q122

Which one of the following equations is a correct identity for arbitrary 3 × 3 real matrices P, Q and R?

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Q123

The rank of the matrix

\left[ {\begin{array}{*{20}{c}} 1&1&1 \\ 1&{ - 1}&0 \\ 1&1&1 \end{array}} \right]

is

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Q124

It is given that X₁, X₂, ... X_M are M non-zero, orthogonal vectors. The dimension of the vector space spanned by the 2M vectors X₁, X₂ ... X_M, -X₁, -X₂ ... -X_M is

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Q125

Consider the set of (column) vectors defined by X = {x

\in

R³ | x₁ + x₂ + x₃ = 0, where x^T =[x₁, x₂, x₃]^T}. Which of the following is TRUE?

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Q126

For given matrix

{\text{P}} = \left[ {\begin{array}{*{20}{c}} {4 + 3{\text{i}}}&{ - {\text{i}}} \\ {\text{i}}&{4 - 3{\text{i}}} \end{array}} \right]

where

i = - 1,

the inverse of matrix P is

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Q127

The matrix

\left[ {\begin{array}{*{20}{c}} 1&2&4 \\ 3&0&6 \\ 1&1&{\text{p}} \end{array}} \right]

has one eigen value equal to 3. The sum of the other two eigen values is

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Q128

The linear operation L(x) is defined by the cross product L(x) = b × X, where b = [0 1 0]^T and X = [x₁x₂x₃]^T are three dimensional vectors. The 3 × 3 matrix M of this operation satisfies

{\text{L}}\left( {\text{x}} \right) = {\text{M}}\left[ {\begin{array}{*{20}{c}} {{{\text{x}}_1}} \\ {{{\text{x}}_2}} \\ {{{\text{x}}_3}} \end{array}} \right].

Then the eigen values of M are

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Q129

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?

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Q130

The rank of the following matrix is

\left( {\begin{array}{*{20}{c}} 1&1&0&{ - 2} \\ 2&0&2&2 \\ 4&1&3&1 \end{array}} \right)

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Q131

The transformation matrix for mirroring a point in x-y plane about the line y = x is given by

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Q132

For the matrix,

\left[ {\begin{array}{*{20}{c}} 4&1 \\ 1&4 \end{array}} \right]

the eigen values are

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Q133

For the matrix A satisfying the equation given below, the eigen values are

\left[ {\text{A}} \right]\left[ {\begin{array}{*{20}{c}} 1&2&3 \\ 7&8&9 \\ 4&5&6 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&2&3 \\ 4&5&6 \\ 7&8&9 \end{array}} \right]

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Q134

The matrix

{\text{M}} = \left[ {\begin{array}{*{20}{c}} { - 2}&2&{ - 3} \\ 2&1&{ - 6} \\ { - 1}&{ - 2}&0 \end{array}} \right]

has eigen values -3, -3, 5. An eigen vector corresponding to the eigen value 5 is [1 2 -1]^T. One of the eigen vectors of the matrix M³ is

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Q135

The inverse of the matrix

\left[ {\begin{array}{*{20}{c}} {3 + 2{\text{i}}}&{\text{i}} \\ { - {\text{i}}}&{3 - 2{\text{i}}} \end{array}} \right]

is

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Q136

In matrix equation [A]{X} = {R}

\left[ {\text{A}} \right] = \left[ {\begin{array}{*{20}{c}} 4&8&4 \\ 8&{16}&{ - 4} \\ 4&{ - 4}&{15} \end{array}} \right],\,\left\{ {\text{X}} \right\} = \left\{ {\begin{array}{*{20}{c}} 2 \\ 1 \\ 4 \end{array}} \right\}\,{\text{and }}\left\{ {\text{R}} \right\} = \left\{ {\begin{array}{*{20}{c}} {32} \\ {16} \\ {64} \end{array}} \right\}

One of the eigen values of Matrix [A] is

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Q137

Consider a 2 × 2 square matrix

{\text{A}} = \left[ {\begin{array}{*{20}{c}} \sigma &{\text{x}} \\ \omega &\sigma \end{array}} \right]

where x is unknown. If the eigen values of the matrix A are

(σ + j ω)

and

(σ - j ω)

, then x is equal to

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Q138

Solution for the system defined by the set of equations 4y + 3z = 8; 2x - z = 2 and 3x + 2y = 5 is

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Q139

The solution to the system of equations is

\left[ {\begin{array}{*{20}{c}} 2&5 \\ { - 4}&3 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\text{x}} \\ {\text{y}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2 \\ { - 30} \end{array}} \right]

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Q140

For the given orthogonal matrix Q

{\text{Q}} = \left[ {\begin{array}{*{20}{c}} {\frac{3}{7}}&{\frac{2}{7}}&{\frac{6}{7}} \\ { - \frac{6}{7}}&{\frac{3}{7}}&{\frac{2}{7}} \\ {\frac{2}{7}}&{\frac{6}{7}}&{ - \frac{3}{7}} \end{array}} \right]

The inverse is

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