1
GATE CE 2005
+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 $$i.$$
2
GATE CE 2004
+1
-0.3
Real matrices $$\,\,{\left[ A \right]_{3x1,}}$$ $$\,\,{\left[ B \right]_{3x3,}}$$ $$\,\,{\left[ C \right]_{3x5,}}$$ $$\,\,{\left[ D \right]_{5x3,}}$$ $$\,\,{\left[ E \right]_{5x5,}}$$ $$\,\,{\left[ F \right]_{5x1,}}$$ are given. Matrices $$\left[ B \right]$$ and $$\left[ E \right]$$ are symmetric. Following statements are made with respect to their matrices.
$$(I)$$ Matrix product $$\,\,{\left[ F \right]^T}\,\,$$ $$\,\,{\left[ C \right]^T}\,\,$$ $$\,\,\left[ B \right]\,\,$$ $$\,\,\left[ C \right]\,\,$$ $$\,\,\left[ F \right]\,\,$$ is a scalar.
$$(II)$$ Matrix product $$\,\,{\left[ D \right]^T}\,\,$$ $$\,\left[ F \right]\,\,$$ $$\,\left[ D \right]\,\,$$ is always symmetric.
With reference to above statements which of the following applies?
A
statement $$(I)$$ is true but $$(II)$$ is false
B
statement $$(I)$$ is false but $$(II)$$ is true
C
both the statements are true
D
both the statements are false
3
GATE CE 2004
+1
-0.3
The eigen values of the matrix $$\left[ {\matrix{ 4 & { - 2} \cr { - 2} & 1 \cr } } \right]$$ are
A
$$1,4$$
B
$$-1,2$$
C
$$0.5$$
D
can not be determined
4
GATE CE 2003
+1
-0.3
Given matrix $$\left[ A \right] = \left[ {\matrix{ 4 & 2 & 1 & 3 \cr 6 & 3 & 4 & 7 \cr 2 & 1 & 0 & 1 \cr } } \right],$$ the rank of the matrix is
A
$$4$$
B
$$3$$
C
$$2$$
D
$$1$$
GATE CE Subjects
Construction Material and Management
Geomatics Engineering Or Surveying
Engineering Mechanics
Transportation Engineering
Environmental Engineering
Geotechnical Engineering
Fluid Mechanics and Hydraulic Machines
General Aptitude
EXAM MAP
Medical
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