Problem 19 - College Algebra with Modeling and Visualization

Question

If possible, find each of the following. (a) \(A+B\) (b) \(3 A\) (c) \(2 A-3 B\) $$A=\left[\begin{array}{rr}-2 & -1 \\\\-5 & 1 \\\2 & -3\end{array}\right]$$ $$B=\left[\begin{array}{rr}2 & -1 \\\3 & 1 \\\7 & -5\end{array}\right]$$

Step-by-Step Solution

Answer

(a) \( \left[\begin{array}{rr}0 & -2 \\ -2 & 2 \\ 9 & -8\end{array}\right] \) (b) \( \left[\begin{array}{rr}-6 & -3 \\ -15 & 3 \\ 6 & -9\end{array}\right] \) (c) \( \left[\begin{array}{rr}-10 & 1 \\ -19 & -1 \\ -17 & 9\end{array}\right] \)

Step 1: Determine if matrices can be added

To add matrices, they must have the same dimensions. Matrix \(A\) and matrix \(B\) both have dimensions of \(3 \times 2\), which means they can be added together.

Step 2: Add matrices A and B

Add corresponding elements in matrices \(A\) and \(B\). For example, in the first row and first column, add \(-2 + 2 = 0\). Continue this process for each element:\[A + B = \left[\begin{array}{rr}-2 + 2 & -1 - 1 \ -5 + 3 & 1 + 1 \ 2 + 7 & -3 - 5\end{array}\right] = \left[\begin{array}{rr}0 & -2 \ -2 & 2 \ 9 & -8\end{array}\right]\]

Step 3: Define scalar multiplication for matrix A

Scalar multiplication involves multiplying every element of the matrix by the scalar. For \(3A\), multiply each element of matrix \(A\) by 3. For example, multiply \(-2\) by 3 to get \(-6\):\[3A = 3 \times \left[\begin{array}{rr}-2 & -1 \ -5 & 1 \ 2 & -3\end{array}\right] = \left[\begin{array}{rr}-6 & -3 \ -15 & 3 \ 6 & -9\end{array}\right]\]

Step 4: Calculate 2A and 3B

First, calculate \(2A\) by multiplying each element of matrix \(A\) by 2:\[2A = 2 \times \left[\begin{array}{rr}-2 & -1 \ -5 & 1 \ 2 & -3\end{array}\right] = \left[\begin{array}{rr}-4 & -2 \ -10 & 2 \ 4 & -6\end{array}\right]\]Next, calculate \(3B\) by multiplying each element of matrix \(B\) by 3:\[3B = 3 \times \left[\begin{array}{rr}2 & -1 \ 3 & 1 \ 7 & -5\end{array}\right] = \left[\begin{array}{rr}6 & -3 \ 9 & 3 \ 21 & -15\end{array}\right]\]

Step 5: Subtract 3B from 2A

Subtract corresponding elements in \(3B\) from those in \(2A\). For example, in the first row and first column, subtract \(6\) from \(-4\) to get \(-10\). The process is as follows:\[2A - 3B = \left[\begin{array}{rr}-4 - 6 & -2 + 3 \ -10 - 9 & 2 - 3 \ 4 - 21 & -6 + 15\end{array}\right] = \left[\begin{array}{rr}-10 & 1 \ -19 & -1 \ -17 & 9\end{array}\right]\]

Key Concepts

Matrix Addition

Matrix addition is a simple operation, but there are certain rules and steps to follow to ensure you do it correctly. For two matrices to be added, they must have the same dimensions. This means that if matrix \( A \) is a \( 3 \times 2 \) matrix, then matrix \( B \) must also be \( 3 \times 2 \). In the original exercise, both matrices \( A \) and \( B \) satisfy this condition with dimensions \( 3 \times 2 \).
When adding matrices, you simply add the corresponding elements from each matrix to get a new matrix. Let's look at how this works:

• First element of the result is the first element of \( A \) plus the first element of \( B \): \( -2 + 2 = 0 \).
• Second element becomes: \( -1 + (-1) = -2 \).

Continue this way for all other elements.
The result for this operation in the example is: \[A + B = \begin{bmatrix} 0 & -2 \ -2 & 2 \ 9 & -8 \end{bmatrix} \]Understanding the dimensions and correctly matching every element is key to successfully adding matrices.

Scalar Multiplication

Scalar multiplication involves taking each element of a matrix and multiplying it by a scalar, or numerical factor. In the context of matrices, a scalar is a simple number. In the example exercise, we perform scalar multiplication with the matrix \( A \) and the number 3.
Here's how it works:

• Multiply each element in matrix \( A \) by 3.
• For example: \( -2 \times 3 = -6 \).
• Do this for all the elements of matrix \( A \).

After multiplying each element of \( A \) by 3, you get:\[3A = \begin{bmatrix} -6 & -3 \ -15 & 3 \ 6 & -9 \end{bmatrix} \]Scalar multiplication is straightforward and allows you to scale a matrix without changing its structure. It's crucial to work carefully and verify each multiplication step to ensure accuracy.

Matrix Subtraction

Matrix subtraction is very similar to matrix addition, but instead of adding corresponding elements, you subtract them. To perform matrix subtraction, the two matrices must also have the same dimensions, just like with addition.
In our original exercise, we work with matrices after performing scalar multiplications, specifically \( 2A \) and \( 3B \).
The process of matrix subtraction follows these steps:

• Identify corresponding elements in each matrix.
• Subtract the element of the second matrix from the element of the first matrix.
• For example, first element calculation: \(-4 - 6 = -10\).
• Perform the same subtraction for all related elements in your matrices.

The result from the example's subtraction is:\[2A - 3B = \begin{bmatrix} -10 & 1 \ -19 & -1 \ -17 & 9 \end{bmatrix} \]Using these methods, ensure that each step is done carefully to produce a correct final answer. Doing so will help maintain accuracy in your matrix operations.