To solve this problem, we need to identify why Ramone's matrix equation does not have a solution.
The equation is:
[tex]\[ A + \begin{bmatrix} 8 \\ -12 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 & 18 & -21 \end{bmatrix} \][/tex]
When performing matrix addition, the matrices involved must have the same dimensions. In this equation, [tex]\( \begin{bmatrix} 8 \\ -12 \\ 3 \end{bmatrix} \)[/tex] is a column vector of dimensions [tex]\( 3 \times 1 \)[/tex], while [tex]\( \begin{bmatrix} 0 & 18 & -21 \end{bmatrix} \)[/tex] is a row vector of dimensions [tex]\( 1 \times 3 \)[/tex].
The intrinsic property of matrix addition states that it is only defined for matrices of the same order (i.e., having the same number of rows and columns). Here, we are trying to add a [tex]\( 3 \times 1 \)[/tex] matrix to a [tex]\( 1 \times 3 \)[/tex] matrix, which is mathematically invalid because they do not have the same dimensions.
Therefore, the correct explanation for why Ramone's equation has no solution is:
"The matrices must be of the same size in order to perform addition or subtraction."
Thus, the answer is:
[tex]\[ \boxed{2} \][/tex]
