1
GATE ECE 2014 Set 2
Numerical
+1
-0
The maximum value of the determinant among all $$2 \times 2$$ real symmetric matrices with trace $$14$$ is ______.
2
GATE ECE 2014 Set 1
+1
-0.3
For matrices of same dimension $$M,N$$ and scalar $$c,$$ which one of these properties DOES NOT ALWAYS hold ?
A
$${\left( {{M^T}} \right)^T} = M$$
B
$${\left( {cM} \right)^T} = c{\left( M \right)^T}$$
C
$${\left( {M + N} \right)^T} = {M^T} + {N^T}$$
D
$$MN=NM$$
3
GATE ECE 2014 Set 1
Numerical
+1
-0
Consider the matrix $${J_6} = \left[ {\matrix{ 0 & 0 & 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1 & 0 & 0 \cr 0 & 0 & 1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 & 0 & 0 \cr 1 & 0 & 0 & 0 & 0 & 0 \cr } } \right]$$

Which is obtained by reversing the order of the columns of the identity matrix $${{\rm I}_6}$$. Let $$P = {{\rm I}_6} + \alpha \,\,{J_6},$$ where $$\alpha$$ is a non $$-$$ negative real number. The value of $$\alpha$$ for which det $$(P)=0$$ is _______.

4
GATE ECE 2014 Set 1
Numerical
+1
-0
$$A$$ real $$\left( {4\,\, \times \,\,4} \right)$$ matrix $$A$$ satisfies the equation $${A^2} = {\rm I},$$ where $${\rm I}$$ is the $$\left( {4\,\, \times \,\,4} \right)$$ identity matrix. The positive eigen value of $$A$$ is _______.