Problem 27 - College Algebra

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. $$ A=\left[\begin{array}{rr}{2} & {-5} \\ {6} & {7}\end{array}\right], B=\left[\begin{array}{cc}{-9} & {6} \\ {-4} & {2}\end{array}\right], C=\left[\begin{array}{cc}{0} & {9} \\ {7} & {1}\end{array}\right], D=\left[\begin{array}{rrr}{-8} & {7} & {-5} \\ {4} & {3} & {2} \\ {0} & {9} & {2}\end{array}\right], E=\left[\begin{array}{rrr}{4} & {5} & {3} \\ {7} & {-6} & {-5} \\ {1} & {0} & {9}\end{array}\right] $$ $$ 3 D+4 E $$

Step-by-Step Solution

Answer

The result is a 3x3 matrix: \( \begin{bmatrix} -8 & 41 & -3 \\ 40 & -15 & -14 \\ 4 & 27 & 42 \end{bmatrix} \).

Step 1: Identify the Matrices

Matrix \( D \) is a 3x3 matrix and matrix \( E \) is also a 3x3 matrix. Since both matrices have the same dimensions, we can add and scale them.

Step 2: Scale Matrix D by 3

Multiply each element of matrix \( D \) by 3:\[3D = \begin{bmatrix} -8 \times 3 & 7 \times 3 & -5 \times 3 \ 4 \times 3 & 3 \times 3 & 2 \times 3 \ 0 \times 3 & 9 \times 3 & 2 \times 3 \end{bmatrix} = \begin{bmatrix} -24 & 21 & -15 \ 12 & 9 & 6 \ 0 & 27 & 6 \end{bmatrix}\]

Step 3: Scale Matrix E by 4

Multiply each element of matrix \( E \) by 4:\[4E = \begin{bmatrix} 4 \times 4 & 5 \times 4 & 3 \times 4 \ 7 \times 4 & -6 \times 4 & -5 \times 4 \ 1 \times 4 & 0 \times 4 & 9 \times 4 \end{bmatrix} = \begin{bmatrix} 16 & 20 & 12 \ 28 & -24 & -20 \ 4 & 0 & 36 \end{bmatrix}\]

Step 4: Add Scaled Matrices 3D and 4E

Add the corresponding elements of matrices \( 3D \) and \( 4E \):\[3D + 4E = \begin{bmatrix} -24 + 16 & 21 + 20 & -15 + 12 \ 12 + 28 & 9 - 24 & 6 - 20 \ 0 + 4 & 27 + 0 & 6 + 36 \end{bmatrix} = \begin{bmatrix} -8 & 41 & -3 \ 40 & -15 & -14 \ 4 & 27 & 42 \end{bmatrix}\]

Step 5: Present the Final Result

The result of the operation \( 3D + 4E \) is:\[\begin{bmatrix} -8 & 41 & -3 \ 40 & -15 & -14 \ 4 & 27 & 42 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra, which involves combining two matrices of the same dimensions by adding their corresponding elements. To successfully perform matrix addition, both matrices must have identical dimensions. This requirement ensures that each element in the first matrix has a matching element in the second matrix.

• For matrices to be added, they must have the same number of rows and columns.
• Addition is performed element-wise, meaning you add each corresponding element from the matrices.
• The sum of two matrices preserves the dimensions from the original matrices.

In the original exercise, we see the addition of two 3x3 matrices, which results in another 3x3 matrix, demonstrating how matrix addition preserves the size of the matrices involved. It's a straightforward process that lays the groundwork for more complex operations in linear algebra.

Matrix Scaling

Matrix scaling is an operation that involves multiplying each element of a matrix by a constant, known as a scalar. This operation transforms all elements of the matrix proportionally according to the scalar value.

• Scaling does not change the dimensions of the matrix; it only changes the values within the matrix.
• A positive scalar enlarges the elements, while a negative scalar reverses their signs and scales them.
• Scaling by one leaves the matrix unchanged, and scaling by zero results in a matrix of zeros.

In the original exercise, matrices are scaled by different scalars: 3 for matrix \( D \) and 4 for matrix \( E \). This demonstrates how each element in the matrices is affected by the respective scalar.

3x3 Matrices

A 3x3 matrix is a square matrix that consists of three rows and three columns. Such matrices are common in various fields, including physics, computer graphics, and machine learning, due to their ability to represent transformations and more complex structures.

• 3x3 matrices are capable of storing nine numerical values.
• They are often used in operations like matrix addition, scaling, and multiplication.
• The proper handling of these matrices is crucial for solving real-world problems that require multi-dimensional data representation.

This simplicity, combined with their ability to represent complex operations, makes 3x3 matrices an excellent educational tool for understanding matrix operations. They provide ideal practice scenarios like the exercise with matrices \( D \) and \( E \).

Linear Algebra

Linear algebra is a branch of mathematics devoted to understanding vector spaces and linear mappings between these spaces. It's a foundational subject that underpins many areas of mathematics and applied sciences, providing tools for analyzing and constructing systems where linear transformations are key.

• Linear algebra focuses on vectors, spaces, and transformations.
• Matrices are a fundamental part of linear algebra, representing linear transformations and systems of linear equations.
• Understanding operations like matrix addition and scaling are essential in mastering linear algebra, as illustrated in the exercise above.

Mastery of linear algebra concepts equips students with the analytical skills to work with complex systems, predict dynamic behaviors, and solve real-world problems involving multi-dimensional data. The ability to perform matrix operations such as those illustrated in the problem lays a vital groundwork for tackling more advanced mathematical challenges.