Homogeneous Linear Systems

Q. The non zero value of K such that the system of equations,
$x+Ky+3z=0,4x+3y+Kz=0,2x+y+2z=0$
has non-trival solution is

1. 4
2. 2.5
3. 3.5
4. None

Q.

What is Cramers rule in determinants?

Q. For the following set of simultaneous equations

1.5x - 0.5y = 2

4x + 2y + 3z = 9

7x + y + 5z = 10

1. The solution is unique
2. Infinitely many solutions exist
3. The equations are incompatible
4. Finite number of multiple solutions exist

Q. The number of linearly independent solutions of the system of equations

$\left[\begin{array}{ccc}1& 0& 2\\ 1& -1& 0\\ 2& -2& 0\end{array}\right]\left[\begin{array}{c}{X}_1\\ X_2\\ X_3\end{array}\right]=0$ is equal to

1. 1
2. 2
3. 3
4. 0

Q. A system of equations represented by AX = 0, where X is a column vector of unknown and A is a matrix containing coefficient has non-trivial solution when A is

1. Non-singular
2. Singular
3. Symmetric
4. Hermitian

Q. The value of q, for which the following set of linear equations 2x + 3y = 0, 6x + qy = 0 can have non - trivial solution is

1. 2
2. 7
3. 9
4. 11

Q. $LetP=\left[\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right],abc=1anda,b,c,ϵRandX={\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]}_{3\times 1};thenPX=0$
has infinitely many solution if trace (P) is _________.

1. 3

Q. For _______ value of $K\in (0,\frac{3}{2})$ does the following system of equation admits a non-trivial solution.

x + Ky + 3z = 0;
Kx + 2y + 2z = 0;
2x + 3y + 4z =0

1. 1.25

Q. Consider matrix $A=\left[\begin{array}{ccc}k& 2k& \\ k^2-k& k^2& \end{array}\right]$and vector x=$\left[\begin{array}{cc}{x}_1& \\ x_2& \end{array}\right]$. The number of distinct real values of k for which the equation Ax=0 has infinitely many solutions is ______ .

1. 2