Questions & Answers of Systems of linear equations

Weightage of Systems of linear equations

Total 4 Questions have been asked from Systems of linear equations topic of Linear Algebra subject in previous GATE papers. Average marks 1.25.

Let c1,....,cn be scalars, not all zero, such that $\style{font-family:'Times New Roman'}{\sum_\limits{i=1}^nc_ia_i=0}$ where ai are column vectors in Rn Consider the set of linear equations
                                                      $\style{font-family:'Times New Roman'}{Ax=b}$
where $\style{font-family:'Times New Roman'}{A=\left[a_1,....,a_n\right]\;\mathrm{and}\;b=\sum_\limits{i=1}^na_i}$. The set of equations has 

Consider the following system of equations:

3x + 2y = 1
4x + 7z = 1
x + y + z = 3
x – 2y + 7z = 0

The number of solutions for this system is __________________

The following system of equations

x1+x2+2 x3 = 1
 x1+2 x2+3 x3 = 2
x1+4 x2+
α x3 = 4

has a unique solution. The only possible value(s) for α is/are

Consider the set of (column) vectors defined by X=xR3|x1+x2+x3=0, where xT=x1,x2,x3T. Which of the following is TRUE?