1
GATE CSE 2011
+2
-0.6
Four matrices $${M_1},\,\,\,{M_2},\,\,\,{M_3}$$ and $${M_4}$$ of dimensions $$p\,\,x\,\,q,\,\,\,\,\,q\,\,x\,\,e,\,\,\,\,\,r\,\,x\,\,s$$ and $$\,\,\,\,s\,\,x\,\,t$$ respectively can be multiplied in sevaral ways with different number of total scalar multiplications. For example when multiplied as $$\left( {\left( {{M_1}\,\,X\,\,{M_2}} \right)\,\,X\,\,\left( {{M_3}\,\,X\,\,{M_4}} \right)} \right)$$, the total number of scalar multiplications is $$\,\,\,\,$$$$pqr + rst + prt$$. When multiplied as $$\left( {\left( {\left( {{M_1}\,\,X\,\,{M_2}} \right)\,\,X\,\,{M_3}} \right)X\,\,{M_4}} \right)$$, the total number of scalar multiplications is $$pqr + prs + pst$$. If $$p = 10,\,\,q = 100,\,\,r = 20,\,\,s = 5,\,\,$$ and $$t = 80$$, then the minimum number of scalar multiplications needed is
A
$$248000$$
B
$$44000$$
C
$$19000$$
D
$$25000$$
2
GATE CSE 2011
+2
-0.6
Consider the matrix as given below. $$\left[ {\matrix{ 1 & 2 & 3 \cr 0 & 4 & 7 \cr 0 & 0 & 3 \cr } } \right]$$\$

Which of the following options provides the Correct values of the Eigen values of the matrix?

A
$$1, 4, 3$$
B
$$3, 7, 3$$
C
$$7, 3, 2$$
D
$$1, 2, 3$$
3
GATE CSE 2011
+2
-0.6
$$\left[ A \right]$$ is a square matrix which is neither symmetric nor skew-symmetric and $${\left[ A \right]^T}$$ is its transpose. The sum and differences of these matrices and defined as $$\left[ S \right] = \left[ A \right] + {\left[ A \right]^T}$$ and $$\left[ D \right] = \left[ A \right] - {\left[ A \right]^T}$$ respectively. Which of the following statements is true?
A
Both $$\left[ S \right]$$ and $$\left[ D \right]$$ are symmetric
B
Both $$\left[ S \right]$$ and $$\left[ D \right]$$ are skew-symmetric
C
$$\left[ S \right]$$ is skew-symmetric and $$\left[ D \right]$$ is symmetric
D
$$\left[ S \right]$$ is symmetric and $$\left[ D \right]$$ is skew symmetric
4
GATE CSE 2011
+2
-0.6
Consider a finite sequence of random values $$X = \left\{ {{x_1},{x_2},{x_3}, - - - - - {x_n}} \right\}..$$ Let $${\mu _x}$$ be the mean and $${\sigma _x}$$ be the standard deviation of $$X.$$ Let another finite sequence $$Y$$ of equal length be derived from this $${y_i} = a.{x_i} + b,$$ where $$a$$ and $$b$$ are positive constants. Let $${\mu _y}$$ be the mean and $${\sigma _y}$$ be the standard deviation of this sequence. Which one of the following statements is incorrect?
A
Index position of mode of $$X$$ in $$X$$ is the same as the index position of mode of $$Y$$ in $$Y$$.
B
Index position of median of $$X$$ in $$X$$ is the same as the index position of median of $$Y$$ in $$Y.$$
C
$${\mu _y} = a{\mu _x} + b$$
D
$${\sigma _y} = a{\sigma _x} + b$$
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