1
GATE CE 2005
MCQ (Single Correct Answer)
+1
-0.3
Consider a non-homogeneous system of linear equations represents mathematically an over determined system. Such a system will be
A
Consistent having a unique solution
B
Consistent having many solutions.
C
Inconsistent having a unique solution.
D
none
2
GATE CE 2005
MCQ (Single Correct Answer)
+1
-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

A
no solution
B
a unique solution
C
more then one but a finite number of solutions.
D
an infinite number of solutions.
3
GATE CE 2005
MCQ (Single Correct Answer)
+1
-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
A
$$2 \times 2$$
B
$$3 \times 3$$
C
$$\,4 \times 3\,$$
D
$$\,3 \times 4\,$$
4
GATE CE 2005
MCQ (Single Correct Answer)
+1
-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?
A
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 $$n.$$
B
For matrix $${A^m},$$ $$m$$ being a positive integer, $$\left( {{\lambda _i}^m,\,{X_i}^m} \right)$$ will be eigen pair for all $$i.$$
C
If $${A^T} = {A^{ - 1}}$$ then $$\left| {{\lambda _i}} \right| = 1$$ for all $$i.$$
D
If $${A^T} = A$$ then $${{\lambda _i}}$$ are real for all $$i.$$
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