Linear Algebra MCQs Question and Answer in Engineering Maths

If the vectors e₁ = (1, 0, 2), e₂ = (0, 1, 0) and e₃ = (-2, 0, 1) form an orthogonal basis of the three-dimensional real space R³, then the vector u = (4, 3, -3) \[ \in \] R³ can be expressed as

A real n × n matrix A = {a_ij} is defined as follows: a_ij = i, if i = j, otherwise 0
The summation of all n eigen values of A is

The value of q for which the following set of linear equation 2x + 3y = 0; 6x + qy = 0 can have non-trivial solution is

For a matrix \[\left[ {\text{M}} \right] = \left[ {\begin{array}{*{20}{c}} {\frac{3}{5}}&{\frac{4}{5}} \\ {\text{x}}&{\frac{3}{5}} \end{array}} \right],\]    the transpose of the matrix is equal to the inverse of the matrix, [M]^T = [M]^-1. The value of x is given by

Consider a matrix P whose only eigenvectors are the multiples of \[\left[ {\begin{array}{*{20}{c}} 1 \\ 4 \end{array}} \right].\]
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?

Let \[{\text{P}} = \left[ {\begin{array}{*{20}{c}} 3&1 \\ 1&3 \end{array}} \right].\]   Consider the set S of all vectors \[\left( {\begin{array}{*{20}{c}} {\text{x}} \\ {\text{y}} \end{array}} \right)\] such that a² + b² = 1 where \[\left( {\begin{array}{*{20}{c}} {\text{a}} \\ {\text{b}} \end{array}} \right) = {\text{P}}\left( {\begin{array}{*{20}{c}} {\text{x}} \\ {\text{y}} \end{array}} \right).\]   Then S is

In the given matrix \[\left[ {\begin{array}{*{20}{c}} 1&{ - 1}&2 \\ 0&1&0 \\ 1&2&1 \end{array}} \right],\]   one of the eigen values is 1. The eigen vectors corresponding to the eigen value 1 are

Let X be a square matrix. Consider the following two statements on X.
I. X is invertible.
II. Determinant of X is non-zero.
Which one of the following is TRUE?

Choose the CORRECT set of functions, which are linearly dependent.

Let M⁴ = $$I$$, (where $$I$$ denotes the identity matrix) and M ≠ $$I$$, M² ≠ $$I$$ and M³ ≠ $$I$$. Then, for any natural number k, M^-1 equals: