where $$A = \left[ {\matrix{ 1 & 0 & 1 & 0 \cr 0 & 1 & 0 & 1 \cr 1 & 1 & 0 & 0 \cr 0 & 0 & 0 & 1 \cr } } \right]\,\,\& \,\,b = \left[ {\matrix{ 0 \cr 0 \cr 0 \cr 1 \cr } } \right]$$
Which of the following statement is true?
where $$A = \left[ {\matrix{ 1 & 0 & 1 & 0 \cr 0 & 1 & 0 & 1 \cr 1 & 1 & 0 & 0 \cr 0 & 0 & 0 & 1 \cr } } \right]\,\,\& \,\,b = \left[ {\matrix{ 0 \cr 0 \cr 0 \cr 1 \cr } } \right]$$
Then the rank of matrix $$A$$ is
$$A\left[ {\matrix{ 1 \cr { - 2} \cr } } \right] = - 2\left[ {\matrix{ 1 \cr { - 2} \cr } } \right]$$ then the matrix $$A$$ is
Group-$${\rm I}$$
$$P.$$$$\,\,\,\,$$ General solution of Homogeneous equations
$$Q.$$$$\,\,\,\,$$ Particular integral
$$R.$$$$\,\,\,\,$$ Total solution satisfying boundary Conditions
Group-$${\rm II}$$
$$(1)$$$$\,\,\,\,$$ $$0.1\,{e^x}$$
$$(2)$$$$\,\,\,\,$$ $$\,{e^{ - x}}\left[ {A\,\cos \,10x + B\,\sin \,10x} \right]$$
$$(3)$$$$\,\,\,\,$$ $${e^{ - x}}\,\cos \,10x + 0.1\,{e^x}$$
Group-$${\rm I}$$
$$P.$$$$\,\,\,\,$$ General solution of Homogeneous equations
$$Q.$$$$\,\,\,\,$$ Particular integral
$$R.$$$$\,\,\,\,$$ Total solution satisfying boundary Conditions
Group-$${\rm II}$$
$$(1)$$$$\,\,\,\,$$ $$0.1\,{e^x}$$
$$(2)$$$$\,\,\,\,$$ $$\,{e^{ - x}}\left[ {A\,\cos \,10x + B\,\sin \,10x} \right]$$
$$(3)$$$$\,\,\,\,$$ $${e^{ - x}}\,\cos \,10x + 0.1\,{e^x}$$