If [x \ 1 \ 1] pmatrix 1 & 2 & 3 4 & 5 & 6 7 & 8 & 9 pmatrix...

Given equation is $[x \ 1 \ 1] \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ x \end{pmatrix} = [45]$

Multiplying the first two matrices:

$[x(1) + 1(4) + 1(7) \quad x(2) + 1(5) + 1(8) \quad x(3) + 1(6) + 1(9)] \begin{pmatrix} 1 \\ 1 \\ x \end{pmatrix} = [45]$

$[x + 11 \quad 2x + 13 \quad 3x + 15] \begin{pmatrix} 1 \\ 1 \\ x \end{pmatrix} = [45]$

Multiplying the resulting row matrix with the column matrix:

$(x + 11)(1) + (2x + 13)(1) + (3x + 15)(x) = 45$

$x + 11 + 2x + 13 + 3x^2 + 15x = 45$

$3x^2 + 18x + 24 = 45$

$3x^2 + 18x - 21 = 0$

$x^2 + 6x - 7 = 0$

$(x + 7)(x - 1) = 0$

$x = -7$ or $x = 1$

From the given options, $x = 1$ is a valid value.

Answer: $1$