Problem 12 - 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) \(\boldsymbol{B}+\boldsymbol{A} \quad(\boldsymbol{c}) \boldsymbol{A}-\boldsymbol{B}\) $$A=\left[\begin{array}{rr}2 & -4 \\\\-1 & \frac{1}{2} \\\3 & -2\end{array}\right]$$ $$B=\left[\begin{array}{rr}5 & 0 \\\3 & \frac{1}{2} \\\\-1 & 1\end{array}\right]$$

Step-by-Step Solution

Answer

(a) \(A+B = \begin{bmatrix} 7 & -4 \\ 2 & 1 \\ 2 & -1 \end{bmatrix}\), (b) \(B+A = \begin{bmatrix} 7 & -4 \\ 2 & 1 \\ 2 & -1 \end{bmatrix}\), (c) \(A-B = \begin{bmatrix} -3 & -4 \\ -4 & 0 \\ 4 & -3 \end{bmatrix}\).

Step 1: Understanding Matrix Addition

Matrix addition involves adding corresponding elements from both matrices. For matrices \(A\) and \(B\) of the same dimensions, each element of the sum, \( (A+B)_{ij} \), is calculated as \( A_{ij} + B_{ij} \).

Step 2: Add Matrices A and B

Given matrices \(A\) and \(B\), add corresponding elements to compute \(A + B\):\[A+B = \begin{bmatrix} 2+5 & -4+0 \ -1+3 & \frac{1}{2}+\frac{1}{2} \ 3+(-1) & -2+1 \end{bmatrix}= \begin{bmatrix} 7 & -4 \ 2 & 1 \ 2 & -1 \end{bmatrix}\]

Step 3: Verify B + A equals A + B

Matrix addition is commutative, so \( B+A = A+B \). To verify, repeat addition with matrices \( B \) and \( A \), resulting in the same matrix:\[B+A = \begin{bmatrix} 5+2 & 0+(-4) \ 3+(-1) & \frac{1}{2}+\frac{1}{2} \ -1+3 & 1+(-2) \end{bmatrix} = \begin{bmatrix} 7 & -4 \ 2 & 1 \ 2 & -1 \end{bmatrix}\]

Step 4: Understanding Matrix Subtraction

Matrix subtraction is similar to addition but involves subtracting corresponding elements. For \(A-B\), compute each element as \( A_{ij} - B_{ij} \).

Step 5: Subtract Matrix B from A

To find \(A-B\), subtract corresponding elements of \(B\) from \(A\):\[A-B = \begin{bmatrix} 2-5 & -4-0 \ -1-3 & \frac{1}{2}-\frac{1}{2} \ 3-(-1) & -2-1 \end{bmatrix} = \begin{bmatrix} -3 & -4 \ -4 & 0 \ 4 & -3 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra, where corresponding elements in two matrices of the same dimensions are added together. If you have two matrices, say matrix \( A \) and matrix \( B \), the sum \( A + B \) is calculated by adding their elements respectively. For example, if the first element in the first row of \( A \) is 2 and the corresponding element in \( B \) is 5, then their sum in the resulting matrix is \( 2 + 5 = 7 \). Similarly, this is done for all elements of the matrices.

In practice:

• Add elements at position \((i, j)\) from both matrices.
• Ensure both matrices being added have the same dimensions (e.g., 2x3).
• For example, if \( A = \begin{bmatrix} 2 & -4 \ -1 & \frac{1}{2} \ 3 & -2 \end{bmatrix} \) and \( B = \begin{bmatrix} 5 & 0 \ 3 & \frac{1}{2} \ -1 & 1 \end{bmatrix} \), then: \[ A + B = \begin{bmatrix} 2 + 5 & -4 + 0 \ -1 + 3 & \frac{1}{2} + \frac{1}{2} \ 3 + (-1) & -2 + 1 \end{bmatrix} = \begin{bmatrix} 7 & -4 \ 2 & 1 \ 2 & -1 \end{bmatrix} \]

Matrix Subtraction

Matrix subtraction is another important operation where you subtract elements in one matrix from the corresponding elements in another matrix. As with matrix addition, it is essential that the matrices have the same dimensions to be subtracted. For matrices \( A \) and \( B \), the expression \( A - B \) means for each element, you will take the element in \( A \) and subtract the corresponding element in \( B \).

In practice:

• Subtract the element in position \((i, j)\) in \( B \) from the element in the same position in \( A \).
• This process needs the matrices to align size-wise, such as both being 3x3 matrices.
• For example, with matrices \( A = \begin{bmatrix} 2 & -4 \ -1 & \frac{1}{2} \ 3 & -2 \end{bmatrix} \) and \( B = \begin{bmatrix} 5 & 0 \ 3 & \frac{1}{2} \ -1 & 1 \end{bmatrix} \), the subtraction results in: \[ A - B = \begin{bmatrix} 2 - 5 & -4 - 0 \ -1 - 3 & \frac{1}{2} - \frac{1}{2} \ 3 - (-1) & -2 - 1 \end{bmatrix} = \begin{bmatrix} -3 & -4 \ -4 & 0 \ 4 & -3 \end{bmatrix} \]

Commutative Property

The commutative property is a fundamental principle in mathematics stating that the order of operation does not change the result when applying addition. This property holds true for matrix addition as well. Simply put, for any two matrices \( A \) and \( B \), the equation \( A + B = B + A \) will always be true. This property means you can switch the order of the matrices being added and still end up with the same result.

Applications:

• This property is extremely helpful in simplifying complex matrix expressions, allowing for flexibility in computation.
• The commutative nature assures that collaborative mathematic solutions remain consistent regardless of operational order.
• However, it's important to note that the commutative property does not apply to operations like subtraction; \( A - B \) is not the same as \( B - A \).