Problem 27 - College Algebra with Corequisite Support

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. \(A=\left[\begin{array}{rr}2 & -5 \\ 6 & 7\end{array}\right], B=\left[\begin{array}{ll}-9 & 6 \\ -4 & 2\end{array}\right], C=\left[\begin{array}{ll}0 & 9 \\ 7 & 1\end{array}\right], D=\left[\begin{array}{rrr}-8 & 7 & -5 \\ 4 & 3 & 2 \\ 0 & 9 & 2\end{array}\right], E=\left[\begin{array}{rrr}4 & 5 & 3 \\ 7 & -6 & -5 \\ 1 & 0 & 9\end{array}\right]\) \(3 D+4 E\)

Step-by-Step Solution

Answer

The resulting matrix of the operation \(3D + 4E\) is \(\begin{bmatrix} -8 & 41 & -3 \\ 40 & -15 & -14 \\ 4 & 27 & 42 \end{bmatrix}\).

Step 1: Scale Matrix D by 3

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

Step 2: Scale Matrix E by 4

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

Step 3: Add Scaled Matrices

Add the results from Step 1 and Step 2 to find \(3D + 4E\). The matrices from steps 1 and 2 are:\[3D = \begin{bmatrix} -24 & 21 & -15 \ 12 & 9 & 6 \ 0 & 27 & 6 \end{bmatrix}, \quad 4E = \begin{bmatrix} 16 & 20 & 12 \ 28 & -24 & -20 \ 4 & 0 & 36 \end{bmatrix}\]Add corresponding elements:\[3D + 4E = \begin{bmatrix} -24+16 & 21+20 & -15+12 \ 12+28 & 9+(-24) & 6+(-20) \ 0+4 & 27+0 & 6+36 \end{bmatrix} = \begin{bmatrix} -8 & 41 & -3 \ 40 & -15 & -14 \ 4 & 27 & 42 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is one of the fundamental operations in matrix algebra. It involves adding two matrices by adding their corresponding elements. For this operation to be performed, both matrices must have the same dimensions. This means each matrix must have the same number of rows and columns.

When you add matrices, you simply take the first entry from the first matrix and add it to the first entry of the second matrix. Continue this process for each corresponding element.

• If matrix A is 2x3 and matrix B is also 2x3, their corresponding elements can be added.
• Resulting matrix will retain the same dimension, in this case, 2x3.

In the equation given, after scaling matrices 3D and 4E, you align them and add each element from 3D to the matching element in 4E. The operation is complete only if both matrices share the same dimensions.

Scalar Multiplication

In matrix algebra, scalar multiplication involves multiplying each entry of a matrix by a scalar (a real number). Unlike matrix addition, scalar multiplication can be performed on any matrix, irrespective of its dimensions.

Consider multiplying matrix D by scalar 3, which involves:

• Take each entry of matrix D, such as \(-8\), and multiply by 3 to get \(-24\).
• Repeat this for every element in the matrix D.
• The resulting matrix's dimensions remain unchanged, only the values within the matrix are altered.

Similarly, multiplying matrix E by 4 works the same way, modifying each element of matrix E by 4.

The order of operations (scaling followed by adding) is crucial to correctly solve problems involving both scalar multiplication and matrix addition.

Matrix Algebra

Matrix algebra encompasses various actions, including addition, subtraction, and multiplication (both scalar and matrix), to analyze and manipulate matrices. Understanding how these operations work together is essential when working with complex matrix equations.

One key aspect is recognizing when operations like addition are possible based on matrix dimensions. Additionally, combining operations such as first scaling matrices and then adding them, as in the problem \(3D + 4E\), showcases the dynamic nature of matrix algebra.

• Execute one operation at a time: first scalar multiplication, then addition.
• Verify dimensions compatibility before attempting certain operations, like matrix addition.

Through right sequencing and combination of these operations, matrix algebra becomes a powerful tool to solve multidimensional mathematical problems.

Matrix Dimensions

Matrix dimensions are defined by the number of rows and columns in a matrix. They play a critical role in determining whether certain matrix operations, like addition and multiplication, can be performed.

When considering the dimensions, the format "m x n" denotes a matrix with "m" rows and "n" columns. For example, if you have matrix A of dimension 2x2 and matrix B of dimension 3x3, adding them is not possible since they don't share the same dimensions.

In the provided exercise, both matrices 3D and 4E resulting from scaling have dimensions 3x3. This allows their sum to be computed because their dimensions match.

• Check dimensions first before matrix operations to avoid errors.
• Matrix operations are only possible when dimension criteria (like matching in addition) are met.

Understanding matrix dimensions ensures correct application of matrix operations and helps in avoiding operational mistakes in matrix algebra.