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Matrix addition is a fundamental operation in linear algebra. It involves adding two matrices by adding the corresponding entries together. For matrices to be added, they must have the same dimensions. This means both matrices must have the same number of rows and columns. For example, if matrix A and matrix B both have dimensions 2x3, we can add them.
The addition of two matrices A and B, where A = \[[a_{ij}]\] and B = \[[b_{ij}]\], is given by another matrix C = \[[c_{ij}]\] where each element is calculated as:
\[c_{ij} = a_{ij} + b_{ij} \]
Here's an example with simple numbers to illustrate the concept:
Suppose matrix A is:
\[ \begin{array}{rr} 1 & 2 \ 3 & 4 \ \end{array} \] and matrix B is:
\[ \begin{array}{rr} 5 & 6 \ 7 & 8 \ \end{array} \]
Matrix C, the result of A + B, will be:
\[ \begin{array}{rr} 1+5 & 2+6 \ 3+7 & 4+8 \ \end{array} \]
Which simplifies to:
\[ \begin{array}{rr} 6 & 8 \ 10 & 12 \ \end{array} \]
So Matrix C = \[ \begin{array}{rr} 6 & 8 \ 10 & 12 \ \end{array} \]
The same rule applies for more complex matrices, as long as they have the same shape.

Scalar multiplication involves multiplying every element of a matrix by a scalar (a constant number). This operation is straightforward and useful in many matrix operations.
For example, if you have a matrix A:
\[ \begin{array}{rrr} 1 & 2 & 3 \ 4 & 5 & 6 \ \end{array} \]
and you want to multiply it by 3 (scalar multiplication), you calculate 3A as:
\[ \begin{array}{rrr} 3 \times 1 & 3 \times 2 & 3 \times 3 \ 3 \times 4 & 3 \times 5 & 3 \times 6 \ \end{array} \]
Which simplifies to:
\[ \begin{array}{rrr} 3 & 6 & 9 \ 12 & 15 & 18 \ \end{array} \]
Thus, scalar multiplication scales each element of the original matrix by the specified scalar value.
When performing complex operations involving scalar multiplication, like in the given exercise where we need to find 3A and 2B, remember to multiply each element of matrices A and B by 3 and 2, respectively. This step simplifies other operations, such as subtraction, which comes next.

Matrix subtraction is the process of subtracting one matrix from another by subtracting the corresponding entries. For matrix subtraction to be defined, both matrices must have the same dimensions, just like in matrix addition.
Let's consider an example: If Matrix A = \[ \begin{array}{rr} 4 & 7 \ 2 & 5 \ \end{array} \] and Matrix B = \[ \begin{array}{rr} 1 & 3 \ -1 & 2 \ \end{array} \],
Matrix C, the result of A - B, will be:
\[ \begin{array}{rr} 4-1 & 7-3 \ 2+1 & 5-2 \ \end{array} \]
Which simplifies to:
\[ \begin{array}{rr} 3 & 4 \ 3 & 3 \ \end{array} \]
So, Matrix C = \[ \begin{array}{rr} 3 & 4 \ 3 & 3 \ \end{array} \]
In the given exercise, to find 3A - 2B, you would first perform scalar multiplication for both matrices A (multiply by 3) and B (multiply by 2), as shown in the previous section, then subtract the resulting matrices element by element. The final result provides a single matrix that represents 3A - 2B.
This process of combining scalar multiplication and matrix subtraction is key for solving many matrix-related problems in linear algebra.