Consider the system of linear equations as 2x + 2y + z = 1, 4x + ky + 2z = 2 and kx + 4y + z = 1 then choosethe correct statement(s) from blow 

(A) The system of equation has a unique solution if k≠4 and k≠2

(B) The system of equations is inconsistent for every real number k

(C) The system of equations have infinite number of solutions if k = 4

(D) The system of equations have infinite number of solutions if k = 2

Choose the correct answer from the options given below

The system of equations is:

2x + 2y + z = 1
4x + ky + 2z = 2
kx + 4y + z = 1

The coefficient matrix is

\(A = \begin{bmatrix} 2 & 2 & 1 \\ 4 & k & 2 \\ k & 4 & 1 \end{bmatrix}\).

The determinant is

\(\Delta = \begin{vmatrix} 2 & 2 & 1 \\ 4 & k & 2 \\ k & 4 & 1 \end{vmatrix}\).

Expanding:

\(\Delta = 2\begin{vmatrix} k & 2 \\ 4 & 1 \end{vmatrix} - 2\begin{vmatrix} 4 & 2 \\ k & 1 \end{vmatrix} + 1\begin{vmatrix} 4 & k \\ k & 4 \end{vmatrix}\).

\(\Delta = 2(k-8) - 2(4-2k) + (16-k^2)\).

\(\Delta = -k^2 + 6k - 8 = -(k-2)(k-4)\).

• If \(k \neq 2,4\), then \(\Delta \neq 0\) and the system has a unique solution.
• If \(k=4\): equations (1) and (2) are dependent, equation (3) reduces to the same relation, hence infinitely many solutions.
• If \(k=2\): substituting gives \(y=0\) and \(2x+z=1\), equation (3) is the same, hence infinitely many solutions.

Correct Statements: (A), (C), (D)