Problem 14 - College Algebra with Modeling and Visualization

Question

For the given matrices \(A\) and \(B\) find each of the following. (a) \(\boldsymbol{A}+\boldsymbol{B} \quad\) (b) \(\boldsymbol{B}+\boldsymbol{A} \quad(\boldsymbol{c}) \boldsymbol{A}-\boldsymbol{B}\) $$A=\left[\begin{array}{rrrr}1 & 6 & 1 & -2 \\\0 & 1 & 3 & 5 \\\0 & 0 & 1 & -2\end{array}\right]$$ $$B=\left[\begin{array}{rrrr}1 & 0 & 0 & 9 \\\3 & 1 & 0 & 3 \\\\-1 & 4 & 1 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

(a) \(\begin{bmatrix} 2 & 6 & 1 & 7 \\ 3 & 2 & 3 & 8 \\ -1 & 4 & 2 & -4 \end{bmatrix}\), (b) same as (a), (c) \(\begin{bmatrix} 0 & 6 & 1 & -11 \\ -3 & 0 & 3 & 2 \\ 1 & -4 & 0 & 0 \end{bmatrix}\).

Step 1: Confirm Matrix Dimensions

First, let's confirm that matrices \( A \) and \( B \) have the same dimensions. Matrix \( A \) is 3x4 and matrix \( B \) is also 3x4. This means we can perform addition and subtraction operations on them.

Step 2: Add Matrices A and B

To find \( A + B \), add corresponding elements of matrices \( A \) and \( B \). \[A + B = \begin{bmatrix} 1+1 & 6+0 & 1+0 & -2+9 \ 0+3 & 1+1 & 3+0 & 5+3 \ 0-1 & 0+4 & 1+1 & -2-2 \end{bmatrix} = \begin{bmatrix} 2 & 6 & 1 & 7 \ 3 & 2 & 3 & 8 \ -1 & 4 & 2 & -4 \end{bmatrix}\]

Step 3: Add Matrices B and A

Since matrix addition is commutative, \( B + A = A + B \). Therefore, the result is the same as in Step 2: \[ B + A = \begin{bmatrix} 2 & 6 & 1 & 7 \ 3 & 2 & 3 & 8 \ -1 & 4 & 2 & -4 \end{bmatrix} \]

Step 4: Subtract Matrix B from Matrix A

To find \( A - B \), subtract corresponding elements of matrix \( B \) from matrix \( A \).\[A - B = \begin{bmatrix} 1-1 & 6-0 & 1-0 & -2-9 \ 0-3 & 1-1 & 3-0 & 5-3 \ 0+1 & 0-4 & 1-1 & -2+2 \end{bmatrix} = \begin{bmatrix} 0 & 6 & 1 & -11 \ -3 & 0 & 3 & 2 \ 1 & -4 & 0 & 0 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra, where two matrices are combined to create a new matrix. For two matrices to be eligible for addition, they must have identical dimensions. This means that both matrices should have the same number of rows and columns.

If we take Matrix A and Matrix B from the given problem, both are of the size 3x4. Thus, they can be added together.

Matrix addition involves adding the corresponding elements of the two matrices. Concretely, this means we take the element from the first row and first column of Matrix A and add it to the element in the first row and first column of Matrix B, and so forth for every element.

• The sum of elements from the same position in each matrix forms the corresponding entry in the resultant matrix.
• This operation is commutative, meaning that \(A + B = B + A\).

This process effectively combines information from both matrices into one, aiding in broader data manipulation and mathematical computations.

Matrix Subtraction

Matrix subtraction, much like matrix addition, requires that the matrices involved share the same dimensions. When subtracting one matrix from another, we focus on subtracting corresponding elements.

In our example with Matrices A and B, both being 3x4, subtraction is achievable.

Here's the process: one simply subtracts each component in matrix B from the corresponding component in matrix A. It’s essential to carefully follow this rule for each element:

• Locate the elements at the same position within the matrices.
• Subtract the element of Matrix B from the element of Matrix A at these corresponding positions: \(A_{ij} - B_{ij}\).
• The outcome is a new matrix where each position reflects the difference of the initial matrices' elements.

Matrix subtraction helps in determining the change or difference between data sets represented by matrices, making it a valuable tool when analyzing variable quantities or transitions.

Matrix Dimensions

Matrix dimensions refer to the number of rows and columns that a matrix contains, usually expressed as 'm x n', where 'm' is the number of rows, and 'n' is the number of columns.

These dimensions are crucial as they determine the possible operations permitted between matrices, such as addition, subtraction, and multiplication.

For instance, in our problem, both Matrix A and Matrix B are 3x4 matrices.

• Equal dimensions imply that each matrix has the same number of rows and columns, allowing direct element-wise operations like addition or subtraction.
• If matrices are of differing dimensions, operations like addition and subtraction become impossible due to a mismatch in structure.

Thus, always check the dimensions before performing any matrix operation to ensure compatibility. This step is fundamental as it impacts both the feasibility of the operation and the interpretation of its result in real-world scenarios.