The system of linear equations
`x - 2y + z = 4`
`2x + 3y - 4z = 1`
`x - 9y + (2a + 3)z = 5a + 1`
has infinitely many solution for:

Options

A. `a!= 2`
B. `a = 1`
C. No value of a
D. a = 2

Explanation

To determine the value of \( a \) for which the given system of linear equations has infinitely many solutions, we need to analyze the coefficients of the equations. The condition for a system of linear equations to have infinitely many solutions is that the determinant of the coefficient matrix must be zero.

The given equations are:

1. \( x - 2y + z = 4 \)
2. \( 2x + 3y - 4z = 1 \)
3. \( x - 9y + (2a + 3)z = 5a + 1 \)

### Step 1: Write the coefficient matrix
The coefficient matrix \( A \) for the system of equations is:

\[ A = \begin{bmatrix} 1 & -2 & 1 \\ 2 & 3 & -4 \\ 1 & -9 & (2a + 3) \end{bmatrix} \]

### Step 2: Calculate the determinant of the coefficient matrix
To find the determinant of matrix \( A \), we can use the formula for the determinant of a 3x3 matrix:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Where:
- \( a = 1, b = -2, c = 1 \)
- \( d = 2, e = 3, f = -4 \)
- \( g = 1, h = -9, i = (2a + 3) \)

Substituting these values into the determinant formula:

\[ \text{det}(A) = 1 \cdot (3(2a + 3) - (-4)(-9)) - (-2)(2(2a + 3) - (-4)(1)) + 1(2(-9) - 3(1)) \]

### Step 3: Simplify the determinant expression
Calculating each term:

1. First term:
\[ 3(2a + 3) - 36 = 6a + 9 - 36 = 6a - 27 \]

2. Second term:
\[ -2(4a + 6 - 4) = -2(4a + 2) = -8a - 4 \]

3. Third term:
\[ 2(-9) - 3 = -18 - 3 = -21 \]

Putting it all together:
\[ \text{det}(A) = (6a - 27) + (8a + 4) - 21 \]
\[ = 6a - 27 + 8a + 4 - 21 \]
\[ = 14a - 44 \]

### Step 4: Set the determinant to zero
For the system to have infinitely many solutions, we set the determinant equal to zero:

\[ 14a - 44 = 0 \]

### Step 5: Solve for \( a \)
Solving for \( a \):

\[ 14a = 44 \]
\[ a = \frac{44}{14} = \frac{22}{7} \]

### Conclusion
The value of \( a \) for which the system of equations has infinitely many solutions is:

\[ \boxed{\frac{22}{7}} \]