Mathematics Tes...

Which of the following statements is/are true:

I. A X = 0, always have a solution.

2x – 3y = 0 and 2x + αy = 0

For what value of α the system has unique solution.

Consider the below statements -

(i) For a matrix system AX = B, there will be a unique solution only iff |A| ≠ 0.

(ii) A system of equations will always be inconsistent if |A| = 0.

(iii) If |A| = 0 and (adjA)B = 0, then the system will have infinite solutions.

(iv) For consistency |A| will always be non-zero. 

Then which of the following statements are true?

For what value of α, the system of equation: x + y + z = 0, x – y + z = 0 and 2x + 3y + αz = 0 has infinitely many solutions.

A non-singular matrix M = \(\begin{bmatrix} 3 & 6 & 9\\6 & 9 & 12\\9 & 12 & 18\end{bmatrix}\) is such that M^-1 = \(\begin{bmatrix} -2/3 & 0 & 1/3\\0 & 1 & -2/3\\ 1/3& -2/3 & 1/3\end{bmatrix}\) ^. If the system of equations x + 2y + 3z = 5, 2x + 3y + 4z = 7, 3x + 4y + 6z = 11 has a unique solution then xyz = ?

Consider the given system of simultaneous linear equations -

x + ωy + ω²z = 0

ωx + ω²y + z = 0

ω²x + y + ωz = 0

where ω is a complex cube root of unity. Then the system of equations has  

If the system of equations px - 2y - pz = 0, 2x + py - pz = 0, -px + py + 2z = 0 has a non-trivial solution, then the sum of the possible values of p is

For what values of k is the system of equations 2k²x + 3y - 1 = 0, 7x - 2y + 3 = 0, 6kx + y + 1 = 0 consistent?

The given system of equations kx + 5y + 3z = 1, x + 3y + 2z = 3, x + 2y + z = 3 has no solution for -

The system of equations kx + y + z = 1, x + ky + z = k and x + y + kz = k² has no solution if k equals