Problem 13 - 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) \(\boldsymbol{B}+\boldsymbol{A} \quad(\boldsymbol{c}) \boldsymbol{A}-\boldsymbol{B}\) $$A=\left[\begin{array}{rrr}3 & 4 & -1 \\\0 & -3 & 2 \\\\-2 & 5 & 10\end{array}\right]$$ $$B=\left[\begin{array}{rrr}11 & 5 & -2 \\\4 & -7 & 12 \\\6 & 6 & 6\end{array}\right]$$

Step-by-Step Solution

Answer

(a) \( A + B = \begin{bmatrix} 14 & 9 & -3 \\ 4 & -10 & 14 \\ 4 & 11 & 16 \end{bmatrix} \), (b) \( B + A = A + B \), (c) \( A - B = \begin{bmatrix} -8 & -1 & 1 \\ -4 & 4 & -10 \\ -8 & -1 & 4 \end{bmatrix} \).

Step 1: Addition of Matrices A and B

To find \( A + B \), we add each corresponding element of matrices \( A \) and \( B \). Performing the addition:\[ A + B = \begin{bmatrix} 3+11 & 4+5 & -1-2 \ 0+4 & -3-7 & 2+12 \ -2+6 & 5+6 & 10+6 \end{bmatrix} = \begin{bmatrix} 14 & 9 & -3 \ 4 & -10 & 14 \ 4 & 11 & 16 \end{bmatrix} \]

Step 2: Addition of Matrices B and A

Since matrix addition is commutative, the order of addition does not change the result. Hence, \( B + A = A + B \). Therefore,\[ B + A = \begin{bmatrix} 14 & 9 & -3 \ 4 & -10 & 14 \ 4 & 11 & 16 \end{bmatrix} \]

Step 3: Subtraction of Matrix B from A

For \( A - B \), subtract each element of \( B \) from the corresponding element in \( A \). Performing the subtraction:\[ A - B = \begin{bmatrix} 3-11 & 4-5 & -1+2 \ 0-4 & -3+7 & 2-12 \ -2-6 & 5-6 & 10-6 \end{bmatrix} = \begin{bmatrix} -8 & -1 & 1 \ -4 & 4 & -10 \ -8 & -1 & 4 \end{bmatrix} \]

Key Concepts

Matrix Operations

Matrix operations involve arithmetic tasks that can be performed on matrices, such as addition, subtraction, and multiplication. These operations follow specific rules that make them different from regular arithmetic. When dealing with matrix operations, it's important to recognize that matrices are organized in rows and columns.

Matrix addition and subtraction come under the category of element-wise operations. For these operations to be possible, the matrices involved must have the same dimensions, meaning they should have the same number of rows and columns. This ensures that each element from one matrix can correspond directly to an element from the other matrix.

When adding matrices, simply add each element in the first matrix to the corresponding element in the second matrix. For subtraction, subtract each element of the second matrix from the corresponding element in the first matrix. For example, if you have two matrices

• Matrix A: \[ \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \ \end{bmatrix} \]
• Matrix B: \[ \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \ \end{bmatrix} \]

The sum would be:\[\begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} \ a_{21}+b_{21} & a_{22}+b_{22} \\end{bmatrix}\] Understanding these basic operations is crucial for more complex tasks involving matrices.

Commutative Property

The commutative property is a fundamental concept in mathematics, applicable to various operations, including matrix addition. This property states that the order in which you add two numbers does not change the result. In simple terms, \( a + b = b + a \).

This principle also applies to matrix addition. If you have two matrices, \( A \) and \( B \), the sum \( A + B \) will be the same as \( B + A \). This is because you're adding corresponding elements and their initial order doesn't affect the result:

\[\begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \\end{bmatrix} = \begin{bmatrix} b_{11} + a_{11} & b_{12} + a_{12} \ b_{21} + a_{21} & b_{22} + a_{22}\end{bmatrix}\]
It's worth noting that this only applies to addition. Subtraction and multiplication of matrices do not necessarily follow the commutative property. Therefore, it's essential to remember these distinctions when performing matrix operations to avoid mistakes.

Element-wise Operations

Element-wise operations in matrix arithmetic refer to adding, subtracting, or multiplying corresponding elements of two matrices. These operations are foundational to understanding more complex matrix computations.

The most common element-wise operations are addition and subtraction, where you directly apply arithmetic to each pair of corresponding elements from the matrices. For instance, when adding two matrices \( A \) and \( B \) of the same size:

• Find the first element of the resulting matrix by adding the first elements of both matrices \( A \) and \( B \).
• Proceed similarly with the second elements, and so on for the entire matrix.

To subtract one matrix from another, you do almost the same process: subtract each element of the second matrix from the corresponding element of the first. For example, with matrices \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 4 & 3 \ 2 & 1 \end{bmatrix} \), the subtraction \( A - B \) would be:
\[\begin{bmatrix} 1-4 & 2-3 \ 3-2 & 4-1\end{bmatrix}= \begin{bmatrix} -3 & -1 \ 1 & 3\end{bmatrix}\]
These straightforward applications of arithmetic make element-wise operations accessible and essential for anyone studying linear algebra.