Linear Algebra MCQs Question and Answer in Engineering Maths

If $${\left[ {1,\,0,\, - 1} \right]^{\text{T}}}$$  is an eigen vector of the following matrix \[\left[ {\begin{array}{*{20}{c}} 1&{ - 1}&0 \\ { - 1}&2&{ - 1} \\ 0&{ - 1}&1 \end{array}} \right]\]   then corresponding eigen value is

The eigen values of the following matrix are \[\left[ {\begin{array}{*{20}{c}} { - 1}&3&5 \\ { - 3}&{ - 1}&6 \\ 0&0&3 \end{array}} \right]\]

Let A be the 2 × 2 matrix with elements a₁₁ = a₁₂ = a₂₁ = +1 and a₂₂ = -1. Then the eigen values of the matrix A¹⁹ are

How many of the following matrices have an eigen value 1?
\[\left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&0 \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 0&1 \\ 0&0 \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 1&{ - 1} \\ 1&1 \end{array}} \right]{\text{ and }}\left[ {\begin{array}{*{20}{c}} { - 1}&0 \\ 1&{ - 1} \end{array}} \right]\]

A real square matrix A is called skew-symmetric if

If \[{\text{R}} = \left[ {\begin{array}{*{20}{c}} 1&0&{ - 1} \\ 2&1&{ - 1} \\ 2&3&2 \end{array}} \right],\]    then top row of R^-1 is

Let, \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 2&{ - 0.1} \\ 0&3 \end{array}} \right]\]   and \[{{\text{A}}^{ - 1}} = \left[ {\begin{array}{*{20}{c}} {\frac{1}{2}}&{\text{a}} \\ 0&{\text{b}} \end{array}} \right].\]
Then (a + b) = ?

We have a set of 3 linear equations in 3 unknowns. 'X \[ \equiv \] Y' means X and Y are equivalent statements and 'X \[\not \equiv \] Y' means X and Y are not equivalent statements.
P : There is a unique solution.
Q : The equations are linearly independent.
R : All eigen values of the coefficient matrix are nonzero.
S : The determinant of the coefficient matrix is nonzero.
Which one of the following is TRUE?

The elgen values of the matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&5 \\ 0&5&6 \\ 0&{ - 6}&5 \end{array}} \right]\]   are

Consider the following simultaneous equations (with c₁ and c₂ being constants):
3x₁ + 2x₂ = c₁
4x₁ + x₂ = c₂
The characteristics equation for these simultaneous equations is