Given $$ A=\left(\begin{array}{cc} 2 & 1 \\ 3 & -2 \end{array}\right), \quad B=\left(\begin{array}{cc} 4 & -1 \\ -2 & 6 \end{array}\right) $$ find (a) \(3 A\) (b) \(2 B\) (c) \(4 A+3 B\) (d) \(B-2 A\) (e) \(2 A^{\mathrm{T}}\) (f) \((2 A)^{\mathrm{T}}\)

Scalar multiplication is an operation in which each entry of a matrix is multiplied by a fixed number, known as a scalar. The scalar effectively scales each element of the matrix. For example, if we have a matrix \( A \) and a scalar \( k \), the scalar multiplication \( kA \) results in a new matrix where each element \( a_{ij} \) is multiplied by \( k \), giving us \( ka_{ij} \). This process is demonstrated in our textbook exercise examples (a) and (b), where 3 is multiplied with matrix \( A \) and 2 with matrix \( B \) respectively.

Remember, scalar multiplication does not change the dimensions of the original matrix—it only affects the magnitude of its elements. Also, the properties of scalar multiplication include being commutative and distributive over matrix addition, which are handy when solving matrix equations or simplifying expressions involving matrices.

Matrix addition is another fundamental operation where two matrices of the same dimensions are added together element-wise. This means each element of one matrix is added to the corresponding element of the second matrix. For instance, to add two matrices \( A \) and \( B \), you would simply take the element in the first row, first column of \( A \) and add it to the element in the first row, first column of \( B \), then proceed similarly with each subsequent element.

In our exercise, part (c) illustrates matrix addition. After individually multiplying the matrices by their given scalars, the resulting matrices (now of the same dimensions) are added together. Remember, you can only add matrices with the same dimensions; otherwise, the operation is undefined. Make sure the matrices align correctly to avoid errors during addition. The ease of matrix addition makes it a helpful tool in various applications ranging from solving systems of linear equations to transforming geometric figures.

The transpose of a matrix is obtained by swapping the matrix's rows and columns. This operation is denoted by an \( A^{\text{T}} \) for a matrix \( A \). Essentially, the element in the ith row and jth column of the original matrix becomes the element in the jth row and ith column of the transposed matrix. This is showcased in our exercise parts (e) and (f), where we transpose matrix \( A \).

In the context of this problem, transposing after scalar multiplication does not affect the final result, as seen in both examples. This indicates that scalar multiplication commutes with transposition, another important property to keep in mind. Transposing a matrix is a common practice in mathematics and has various applications in areas such as computer graphics, where it's used to manipulate the coordinates of objects in space efficiently.