GATE CE 2005 | Linear Algebra Question 31 | Engineering Mathematics

GATE CE 2005

MCQ (Single Correct Answer)

-0.3

Consider the following system of equations in three real variable $${x_1},$$ $${x_2}$$ and $${x_3}:$$ $$$2{x_1} - {x_2} + 3{x_3} = 1$$$ $$$3{x_1} + 2{x_2} + 5{x_3} = 2$$$ $$$ - {x_1} + 4{x_2} + {x_3} = 3$$$

This system of equations has

no solution

a unique solution

more then one but a finite number of solutions.

an infinite number of solutions.

GATE CE 2005

MCQ (Single Correct Answer)

-0.3

Consider a non-homogeneous system of linear equations represents mathematically an over determined system. Such a system will be

Consistent having a unique solution

Consistent having many solutions.

Inconsistent having a unique solution.

none

GATE CE 2005

MCQ (Single Correct Answer)

-0.3

Consider the matrices $$\,{X_{4x3,}}\,\,{Y_{4x3}}$$ $$\,\,{P_{2x3}}.$$ The order of $$\,{\left[ {P{{\left( {{X^T}Y} \right)}^{ - 1}}{P^T}} \right]^T}$$ will be

$$2 \times 2$$

$$3 \times 3$$

$$\,4 \times 3\,$$

$$\,3 \times 4\,$$

GATE CE 2005

MCQ (Single Correct Answer)

-0.3

Consider the system of equations, $${A_{nxn}}\,\,{X_{nx1}}\,\, = \lambda \,{X_{nx1}}$$ where $$\lambda $$ is a scalar. Let $$\left( {{\lambda _i},\,\,{X_i}} \right)$$ be an eigen value and its corresponding eigen vector for real matrix $$A$$. Let $${{\rm I}_{nxn}}$$ be unit matrix. Which one of the following statement is not correct?

For a homogeneous $$nxn$$ system of linear equations $$\left( {A - \lambda {\rm I}} \right)X = 0,$$ having a non trivial solution, the rank of $$\left( {A - \lambda {\rm I}} \right)$$ is less then $$n.$$

For matrix $${A^m},$$ $$m$$ being a positive integer, $$\left( {{\lambda _i}^m,\,{X_i}^m} \right)$$ will be eigen pair for all $$i.$$

If $${A^T} = {A^{ - 1}}$$ then $$\left| {{\lambda _i}} \right| = 1$$ for all $$i.$$

If $${A^T} = A$$ then $${{\lambda _i}}$$ are real for all $$i.$$

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