Linear Algebra MCQs Question and Answer in Engineering Maths

The two vectors [1 1 1] and [1, a, a²], where \[{\text{a}} = \left( { - \frac{1}{2} + {\text{j}}\frac{{\sqrt 3 }}{2}} \right)\]   , are

Multiplication of matrices E and F is G. Matrices E and G are \[{\text{E}} \equiv \left[ {\begin{array}{*{20}{c}} {\cos \theta }&{ - \sin \theta }&0 \\ {\sin \theta }&{\cos \theta }&0 \\ 0&0&1 \end{array}} \right]{\text{and G}} \equiv \left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right].\]
What is the matrix F?

If a matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 2&4 \\ 1&3 \end{array}} \right]\]  and matrix \[{\text{B}} = \left[ {\begin{array}{*{20}{c}} 4&6 \\ 5&9 \end{array}} \right]\]  the transpose of product of these two matrices i.e., (AB)^T is

Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigen values of this matrix are -1 and 7. What are the values of a and b?
\[{\text{A}} = \left( {\begin{array}{*{20}{c}} 1&4 \\ {\text{b}}&{\text{a}} \end{array}} \right)\]

Consider the following system of equations in three real variables x₁, x₂ and x₃
2x₁ - x₂ + 3x₃ = 1
3x₁ - 2x₂ + 5x₃ = 2
-x₁ - 4x₂ + x₃ = 3
This system of equations has

Consider the matrix
\[{\text{P}} = \left[ {\begin{array}{*{20}{c}} 1&1&0 \\ 0&1&1 \\ 0&0&1 \end{array}} \right]\]
The number of distinct eigen values of P is

The system of linear equations
4x + 2y = 7
2x + y = 6
has

For what value of a, if any, will the following system of equations in x, y and z have a solution?
2x + 3y = 4; x + y + z = 4; x + 2y - z = a

The eigen value of the following matrix \[\left[ {\begin{array}{*{20}{c}} {10}&{ - 4} \\ {18}&{ - 12} \end{array}} \right]\]

Given an orthogonal matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 1&1&1&1 \\ 1&1&{ - 1}&{ - 1} \\ 1&{ - 1}&0&0 \\ 0&0&1&{ - 1} \end{array}} \right],\,{\left[ {{\text{A}}{{\text{A}}^{\text{T}}}} \right]^{ - 1}}\,{\text{is}}\]