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Matrix addition is a fundamental operation where you combine two matrices by adding their corresponding elements. To successfully add matrices, they must be of the same dimensions. For example, if you have matrices \( \boldsymbol{A} \) and \( \boldsymbol{B} \) both with the dimension \(2\times2\), matrix addition involves:

• Taking the element from row 1, column 1 of \( \boldsymbol{A} \) and adding it to the element from row 1, column 1 of \( \boldsymbol{B} \),
• Doing the same for each corresponding position in the two matrices.

In our exercise, after performing the addition of matrices \( \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} \) and \( \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \), we get \( \begin{bmatrix} 1 & 2 \ 1 & 1 \end{bmatrix} \). Remember, matrix addition is commutative, which means \( \boldsymbol{A} + \boldsymbol{B} = \boldsymbol{B} + \boldsymbol{A} \).

Matrix multiplication is a bit more complex than addition. To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. The result will have dimensions equivalent to the number of rows in the first matrix and the number of columns in the second matrix. For our matrices \( \boldsymbol{A} \) and \( \boldsymbol{B} \), the multiplication works by:

• Taking the dot product of the rows of \( \boldsymbol{A} \) with the columns of \( \boldsymbol{B} \).
• This involves matching corresponding elements from each row and column, multiplying them, and summing these products for each position in the resulting matrix.

With our given matrices, when multiplying \( \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} \) and \( \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \), the resulting matrix is \( \begin{bmatrix} 1 & 1 \ 1 & 0 \end{bmatrix} \). This operation is not commutative; generally, \( \boldsymbol{A}\boldsymbol{B} eq \boldsymbol{B}\boldsymbol{A} \).

Matrix powers refer to multiplying a matrix by itself one or more times, much like raising numbers to a power. When you square a matrix \( \boldsymbol{A} \), you multiply it by itself: \( \boldsymbol{A}^{2} = \boldsymbol{A} \times \boldsymbol{A} \). For the matrix \( \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} \), squaring it involves taking the dot product of rows and columns within the matrix, resulting in \( \begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix} \). Matrix powers can reveal insights into the properties of matrices, like stability and transformations in systems represented by the matrices. Interestingly, due to matrix multiplication's nature, the ordering in power computations must remain consistent.

Perhaps one of the most intriguing aspects of matrices is their non-commutative property in multiplication. This means that for matrices \( \boldsymbol{A} \) and \( \boldsymbol{B} \), generally, \( \boldsymbol{A}\boldsymbol{B} eq \boldsymbol{B}\boldsymbol{A} \). This property is crucial, especially in fields such as quantum mechanics or 3D graphics, where the order of transformations (represented by matrices) affects the outcome. In our exercise, the observed differences in expressions, like \((\boldsymbol{A} + \boldsymbol{B})^2\) versus \(\boldsymbol{A}^2 + 2 \boldsymbol{A} \boldsymbol{B} + \boldsymbol{B}^2\), highlight this non-commutative nature. Even if some results align, the paths to these results can vary widely when order changes, showcasing the importance of the matrix order in computations.