Linear Algebra MCQs Question and Answer in Engineering Maths

The eigen values of a symmetric matrix are all

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

The rank of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&1&1 \\ 1&{ - 1}&0 \\ 1&1&1 \end{array}} \right]\]  is

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

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?

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 \[{\text{i}} = \sqrt { - 1} ,\]   the inverse of matrix P is

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

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

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?

The rank of the following matrix is \[\left( {\begin{array}{*{20}{c}} 1&1&0&{ - 2} \\ 2&0&2&2 \\ 4&1&3&1 \end{array}} \right)\]