Is matrix addition commutative for matrices of the same dimensions?

The missing middle column is $\begin{bmatrix} 0 \\ -16 \\ 9 \end{bmatrix}$. This is obtained by applying the transformation A to the middle column of B, which is $\begin{bmatrix} 0 \\ 2 \\ 3 \end{bmatrix}$.

Conditions: A and B are the specific 3x3 matrices shown in the video.; The method of column-wise composition is used.

$A 2\times 2$ transformation matrix maps the standard basis vectors to its own columns. Specifically, the first column of the matrix is the image of the vector $[1, 0]^T$, and the second column is the image of the vector $[0, 1]^T$.

Conditions: The matrix is $2\times 2$.; Working in standard Cartesian coordinates.

Matrix subtraction is performed entrywise. Given two matrices of the same size, subtract the corresponding entry of the second matrix from the corresponding entry of the first matrix to get the result at that position.

Conditions: Both matrices must have identical dimensions.

Each column of $a 2\times 2$ matrix can be read as the coordinates of a vector in the 2D plane. For the matrix $A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$, the first column $\begin{bmatrix} 3 \\ 1 \end{bmatrix}$ represents the vector from the origin to the point (3,1), and the second column $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ represents the vector from the origin to the point (1,2).

Conditions: The matrix is $2\times 2$.; Working in standard Cartesian coordinates.

To compute the matrix for the composition $A \circ B$, apply the transformation $A$ to each column of the matrix $B$. The resulting vectors form the columns of the matrix $A \circ B$.

Conditions: A and B are both 3x3 matrices.; The operation is performed over the real numbers.; These are real linear maps with compatible input and output spaces, represented using standard column-vector coordinates.