How do I solve the following?
[2x1 -3x2 + x3; 4x1 - 2x3] + [x1 +2x2; 0x1 - 2x2; 4x1 + x2]^T
When I do the transpose of the second matrix and try to add them together I get lost. Should I consider x1 +2x2 of matrix 2 as one component when getting its transpose? Or are they two separate components? For example: Would getting the transpose of matrix 2 lead to a new matrix with only 1 row or 2?
Edit: more information
The problem is to compute the linear transformation T(A+C^T)([2;1;1])
We are given matrix A and C $$ A= \begin{bmatrix} 2& -3 &1\\ 4& 0& -2 \end{bmatrix} $$
C=[1 2; 0 -2; 4 1]
I computed the matrix transformation induced by A, Ax, and the matrix transformation induced by C, Cx, as previously shown above.
Note that you found the product $Cx$ incorrectly. In fact, these multiplication is invalid because the dimensions don't match.
Instead, find $C^Tx$ by taking the transpose of $C$ first, then multiplying it by $x$.
The easier approach, however, is to add $A + C^T$ and then multiply the result by $x$ to find $(A + C^T)x$ directly.