1
GATE IN 2015
+1
-0.3
Let $$A$$ be an $$n \times n$$ matrix with rank $$r\left( {0 < r < n} \right).$$ Then $$AX=0$$ has $$p$$ independent solutions, where $$p$$ is
A
$$r$$
B
$$n$$
C
$$n-r$$
D
$$n+r$$
2
GATE IN 2014
+1
-0.3
A scalar valued function is defined as $$f\left( x \right){x^T}Ax + {b^T}x + c,$$ where $$A$$ is a symmetric positive definite matrix with dimension $$n \times n;$$ $$b$$ and $$x$$ are vectors of dimension $$n \times 1$$. The minimum value of $$f(x)$$ will occur when $$x$$ equals.
A
$${\left( {{A^T}A} \right)^{ - 1}}B$$
B
$$- {\left( {{A^T}A} \right)^{ - 1}}B$$
C
$$- \left( {{{{A^{ - 1}}B} \over 2}} \right)$$
D
$${{{A^{ - 1}}B} \over 2}$$
3
GATE IN 2014
+1
-0.3
For the matrix $$A$$ satisfying the equation given below, the eigen values are $$\left[ A \right]\left[ {\matrix{ 1 & 2 & 3 \cr 7 & 8 & 9 \cr 4 & 5 & 6 \cr } } \right] = \left[ {\matrix{ 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 8 & 9 \cr } } \right]$$\$
A
$$(1,-j,j)$$
B
$$(1,1,0)$$
C
$$(1,1,-1)$$
D
$$(1,0,0)$$
4
GATE IN 2013
+1
-0.3
The dimension of the null space of the matrix $$\left[ {\matrix{ 0 & 1 & 1 \cr 1 & { - 1} & 0 \cr { - 1} & 0 & { - 1} \cr } } \right]$$ is
A
$$0$$
B
$$1$$
C
$$2$$
D
$$3$$
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