To determine why Ramone's matrix equation has no solution, we need to examine the requirements for performing matrix addition. The equation provided is:
[tex]\[ A + \begin{bmatrix}
8 \\
-12 \\
3
\end{bmatrix} = \begin{bmatrix}
0 & 18 & -21
\end{bmatrix} \][/tex]
In order to add two matrices, they must have the same dimensions.
Let's analyze the dimensions of each matrix in the equation:
- The matrix [tex]\(\begin{bmatrix}
8 \\
-12 \\
3
\end{bmatrix}\)[/tex] is a column matrix with dimensions [tex]\(3 \times 1\)[/tex].
- The matrix [tex]\(\begin{bmatrix}
0 & 18 & -21
\end{bmatrix}\)[/tex] is a row matrix with dimensions [tex]\(1 \times 3\)[/tex].
Since the matrices [tex]\(\begin{bmatrix}
8 \\
-12 \\
3
\end{bmatrix}\)[/tex] and [tex]\(\begin{bmatrix}
0 & 18 & -21
\end{bmatrix}\)[/tex] do not have the same dimensions, it is impossible to perform matrix addition or subtraction with them. Matrix addition (or subtraction) is defined only when both matrices involved in the operation have the same number of rows and columns.
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.
