1
GATE CSE 2017 Set 2
Numerical
+1
-0
Let $$P = \left[ {\matrix{ 1 & 1 & { - 1} \cr 2 & { - 3} & 4 \cr 3 & { - 2} & 3 \cr } } \right]$$ and $$Q = \left[ {\matrix{ { - 1} & { - 2} & { - 1} \cr 6 & {12} & 6 \cr 5 & {10} & 5 \cr } } \right]$$ be two matrices.
Then the rank of $$P+Q$$ is _______.
2
GATE CSE 2017 Set 1
+1
-0.3
Let $${c_1},.....,\,\,{c_n}$$ be scalars, not all zero, such that $$\sum\limits_{i = 1}^n {{c_i}{a_i} = 0}$$ where $${{a_i}}$$ are column vectors in $${R^{11}}.$$ Consider the set of linear equations $$AX=b$$

Where $$A = \left[ {{a_1},.....,\,\,{a_n}} \right]$$ and $$b = \sum\limits_{i = 1}^n {{a_i}.}$$
The set of equations has

A
a unique solution at $$x\,\,\, = \,\,\,{J_n}$$ where $${J_n}$$ denotes a $$n$$-dimensional vector of all $$1$$
B
no solution
C
infinitely many solutions
D
finitely many solutions
3
GATE CSE 2016 Set 2
+1
-0.3
Consider the system, each consisting of m linear equations in $$n$$ variables.
$$I.$$ $$\,\,\,$$ If $$m < n,$$ then all such system have a solution
$$II.$$ $$\,\,\,$$ If $$m > n,$$ then none of these systems has a solution
$$III.$$ $$\,\,\,$$ If $$m = n,$$ then there exists a system which has a solution

Which one of the following is CORRECT?

A
$$I$$ , $$II$$ and $$III$$ are true
B
Only $$II$$ and $$III$$ are true
C
Only $$III$$ is true
D
None of them is true
4
GATE CSE 2016 Set 2
Numerical
+1
-0
Suppose that the eigen values of matrix $$A$$ are $$1, 2, 4.$$ The determinant of $${\left( {{A^{ - 1}}} \right)^T}$$ is _______.