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Matrix addition is the process of adding two matrices by summing their corresponding elements. This means that you simply take each element from the first matrix and add it to the corresponding element from the second matrix. For example, if we have matrices

• \(A = \begin{bmatrix} 6 & 2 & -3 \end{bmatrix}\)
• \(B = \begin{bmatrix} 4 & -2 & 3 \end{bmatrix}\)

To find \(A + B\), we add each corresponding element:

• \(6 + 4 = 10\)
• \(2 + (-2) = 0\)
• \(-3 + 3 = 0\)

So the result is: \[ A + B = \begin{bmatrix} 10 & 0 & 0 \end{bmatrix}.\]This operation is relatively straightforward and requires that both matrices have the same dimensions.

Matrix subtraction is very similar to matrix addition, but instead you subtract the elements from one matrix from the corresponding elements in the other matrix. The matrices must also have the same dimensions. For the matrices

• \(A = \begin{bmatrix} 6 & 2 & -3 \end{bmatrix}\)
• \(B = \begin{bmatrix} 4 & -2 & 3 \end{bmatrix}\)

We find \(A - B\) by subtracting each corresponding element:

• \(6 - 4 = 2\)
• \(2 - (-2) = 4\)
• \(-3 - 3 = -6\)

Thus, the result is:\[ A - B = \begin{bmatrix} 2 & 4 & -6 \end{bmatrix}.\]Always remember, subtraction involves changing the signs of the elements of the second matrix and then adding.

Scalar multiplication involves multiplying every element of a matrix by a constant (known as the scalar). This operation changes the magnitude of the matrix but not its direction. For a scalar -4 and matrix

• \(A = \begin{bmatrix} 6 & 2 & -3 \end{bmatrix}\)

We calculate for \(-4A\) by multiplying each element in matrix \(A\) by -4:

• \(-4 \times 6 = -24\)
• \(-4 \times 2 = -8\)
• \(-4 \times (-3) = 12\)

Thus, the result is:\[ -4A = \begin{bmatrix} -24 & -8 & 12 \end{bmatrix}.\]This operation scales the size of the matrix while maintaining its original layout of positions.

Linear combinations involve multiplying matrices by scalars and then adding or subtracting the results to form a new matrix. This is very useful in a variety of applications, like solving linear equations or transformations. For example, with matrices

• \(A = \begin{bmatrix} 6 & 2 & -3 \end{bmatrix}\)
• \(B = \begin{bmatrix} 4 & -2 & 3 \end{bmatrix}\)

To find \(3A + 2B\), we follow these steps:1. Multiply each element in matrix \(A\) by 3:

• \(3 \times 6 = 18\)
• \(3 \times 2 = 6\)
• \(3 \times (-3) = -9\)

Resulting in \(3A = \begin{bmatrix} 18 & 6 & -9 \end{bmatrix}\).2. Multiply each element in matrix \(B\) by 2:

• \(2 \times 4 = 8\)
• \(2 \times (-2) = -4\)
• \(2 \times 3 = 6\)

Resulting in \(2B = \begin{bmatrix} 8 & -4 & 6 \end{bmatrix}\).3. Finally, add the results of \(3A\) and \(2B\):

• \(18 + 8 = 26\)
• \(6 + (-4) = 2\)
• \(-9 + 6 = -3\)

Thus, the final result is: \[ 3A + 2B = \begin{bmatrix} 26 & 2 & -3 \end{bmatrix}. \]With linear combinations, you can create new and meaningful transformations by adjusting the coefficients applied to each original matrix.