Problem 30 - Precalculus

Question

perform the indicated operations for each expression, if possible. $$A=\left[\begin{array}{rrr}-1 & 3 & 0 \\\2 & 4 & 1\end{array}\right] \quad B=\left[\begin{array}{rr}0 & 2 & 1 \\\3 & -2 & 4\end{array}\right] \quad C=\left[\begin{array}{rr}0 & 1 \\\2 & -1 \\\3 & 1 \end{array}\right] \quad D=\left[\begin{array}{rr}2 & -3 \\\0 & 1 \\\4 & -2 \end{array}\right]$$ $$E=\left[\begin{array}{rrr}-1 & 0 & 1 \\\2 & 1 & 4 \\\\-3 & 1 & 5 \end{array}\right] \quad F=\left[\begin{array}{r}1 \\\0 \\\\-1\end{array}\right] \quad G=\left[\begin{array}{ll}1 & 2 \\\3 & 4\end{array}\right]$$ $$2A+3E$$

Step-by-Step Solution

Answer

\( \begin{bmatrix} -5 & 6 & 3 \\ 10 & 11 & 14 \end{bmatrix} \).

Step 1: Scalar Multiplication of Matrix A

Multiply each element in matrix \( A \) by 2. \[ 2A = 2 \begin{bmatrix} -1 & 3 & 0 \ 2 & 4 & 1 \end{bmatrix} = \begin{bmatrix} 2 \times -1 & 2 \times 3 & 2 \times 0 \ 2 \times 2 & 2 \times 4 & 2 \times 1 \end{bmatrix} = \begin{bmatrix} -2 & 6 & 0 \ 4 & 8 & 2 \end{bmatrix} \]

Step 2: Scalar Multiplication of Matrix E

Multiply each element in matrix \( E \) by 3. \[ 3E = 3 \begin{bmatrix} -1 & 0 & 1 \ 2 & 1 & 4 \ -3 & 1 & 5 \end{bmatrix} = \begin{bmatrix} 3 \times -1 & 3 \times 0 & 3 \times 1 \ 3 \times 2 & 3 \times 1 & 3 \times 4 \ 3 \times -3 & 3 \times 1 & 3 \times 5 \end{bmatrix} = \begin{bmatrix} -3 & 0 & 3 \ 6 & 3 & 12 \ -9 & 3 & 15 \end{bmatrix} \]

Step 3: Matrix Addition of 2A and 3E

Add the corresponding elements of matrices \( 2A \) and \( 3E \). \[ 2A + 3E = \begin{bmatrix} -2 & 6 & 0 \ 4 & 8 & 2 \end{bmatrix} + \begin{bmatrix} -3 & 0 & 3 \ 6 & 3 & 12 \ -9 & 3 & 15 \end{bmatrix} \] Aligning the addition for clarity (note both are 2x3 matrices): \[ 2A + 3E = \begin{bmatrix} -2 + -3 & 6 + 0 & 0 + 3 \ 4 + 6 & 8 + 3 & 2 + 12 \ -9 & 3 & 15 \end{bmatrix} = \begin{bmatrix} -5 & 6 & 3 \ 10 & 11 & 14 \end{bmatrix} \]

Step 4: Conclusion

The result of the operation \( 2A + 3E \) is a matrix: \( \begin{bmatrix} -5 & 6 & 3 \ 10 & 11 & 14 \end{bmatrix} \).

Key Concepts

Scalar Multiplication

Scalar multiplication in matrices is a process where you multiply every element of a matrix by a constant, known as the scalar. This operation can change the size of numbers in the matrix, but it does not change the structure or shape. For example, if you have the matrix \( A = \begin{bmatrix} -1 & 3 & 0 \ 2 & 4 & 1 \end{bmatrix} \) and you want to multiply it by 2, you would multiply each element in matrix \( A \) by 2:
\(2A = 2 \begin{bmatrix} -1 & 3 & 0 \ 2 & 4 & 1 \end{bmatrix} = \begin{bmatrix} 2 \times -1 & 2 \times 3 & 2 \times 0 \ 2 \times 2 & 2 \times 4 & 2 \times 1 \end{bmatrix} = \begin{bmatrix} -2 & 6 & 0 \ 4 & 8 & 2 \end{bmatrix}\)
This technique is helpful in numerous applications such as adjusting scales in data processing or tweaking parameters in equations.

Matrix Addition

Matrix addition involves adding corresponding elements of two matrices of the same dimension. If the matrices don’t align in size, you can’t add them; they must be the same in the number of rows and columns. Again, let's consider the operation from the exercise \(2A + 3E\) where: \(2A = \begin{bmatrix} -2 & 6 & 0 \ 4 & 8 & 2 \end{bmatrix}\) and \(3E = \begin{bmatrix} -3 & 0 & 3 \ 6 & 3 & 12 \ -9 & 3 & 15 \end{bmatrix} \). Both have dimensions 2x3, thus allowing matrix addition.Notice how each element at the same position in the matrices is summed:

• First row, first column: \((-2) + (-3) = -5\)
• Second row, second column: \(8 + 3 = 11\)
• This process continues for all elements that have corresponding positions.

Unlike scalar multiplication, matrix addition can only occur between matrices with the same dimension, making it less versatile in form.

Matrices

Matrices are rectangular arrays of numbers arranged into rows and columns. They are foundational in mathematics and are used particularly in algebra to solve linear equations, and in computer graphics for transformations.
Each matrix has a defined dimension, expressed in terms of the number of its rows and columns. For example, the matrix \( A \) from our exercise has a dimension of 2x3, meaning 2 rows and 3 columns.
Matrices can store large amounts of data easily and perform many types of arithmetic operations:

• **Scalar Multiplication:** Multiply each element by a constant.
• **Matrix Addition:** Add corresponding entries of two same-sized matrices.
• **Matrix Multiplication:** More complicated, involving dot products.

Understanding matrices is crucial because they provide a structured way of handling complex mathematical operations and facilitate the representation of practical problems.