Step-by-step Solution:

1. Expand the Matrix Equation: The given matrix equation is: $[\begin{matrix} a & b & c \end{matrix} ]\left[ {\begin{matrix} 1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7 \\ \end{matrix} } \right] = [\begin{matrix} 0 & 0 & 0 \\ \end{matrix} ]$ Performing the matrix multiplication, we get a 1x3 matrix: $[a(1)+b(8)+c(7) \quad a(9)+b(2)+c(3) \quad a(7)+b(7)+c(7)] = [0 \quad 0 \quad 0]$ This yields a system of three linear equations: (i) $a + 8b + 7c = 0$ (ii) $9a + 2b + 3c = 0$ (iii) $7a + 7b + 7c = 0$

2. Simplify the System of Equations: From equation (iii), we can factor out the common term 7: $7(a + b + c) = 0$ This simplifies to: (iv) $a + b + c = 0$

3. Incorporate the Plane Equation: We are given that the point P(a, b, c) lies on the plane 2x + y + z = 1. This means the coordinates (a, b, c) must satisfy the equation of the plane: (v) $2a + b + c = 1$

4. Solve for a, b, and c: We now have a system of equations, including (iv) and (v): (iv) $a + b + c = 0$ (v) $2a + b + c = 1$ From equation (iv), we can express b + c in terms of a: $b + c = -a$ Now, substitute this expression for b + c into equation (v): $2a + (-a) = 1$ $a = 1$ Substitute a = 1 back into equation (iv): $1 + b + c = 0 \implies b + c = -1$ Now, substitute a = 1 into one of the initial equations, for instance, equation (i): $1 + 8b + 7c = 0 \implies 8b + 7c = -1$ We have a new system of two linear equations for b and c: (A) $b + c = -1$ (B) $8b + 7c = -1$ From (A), we get c = -1 - b. Substitute this into (B): $8b + 7(-1 - b) = -1$ $8b - 7 - 7b = -1$ $b - 7 = -1$ $b = 6$ Finally, find c using c = -1 - b: $c = -1 - 6 = -7$ So, the coordinates of the point P are (a, b, c) = (1, 6, -7).

5. Calculate the Required Value: The question asks for the value of the expression 7a + b + c. Substituting the values we found for a, b, and c: $7a + b + c = 7(1) + (6) + (-7)$ $= 7 + 6 - 7$ $= 6$

6. Conclusion: The value of 7a + b + c is 6. This corresponds to option D.