Problem 20 - A Graphical Approach to College Algebra

Question

Perform each operation if possible. $$\left[\begin{array}{rr} 1 & -4 \\ 2 & -3 \\ -8 & 4 \end{array}\right]-\left[\begin{array}{rr} -6 & 9 \\ -2 & 5 \\ -7 & -12 \end{array}\right]$$

Step-by-Step Solution

Answer

Resulting matrix is \[\begin{bmatrix} 7 & -13 \\ 4 & -8 \\ -1 & 16 \end{bmatrix}\].

Step 1: Verify Matrix Dimensions

Ensure that the matrices involved in the operation have the same dimensions. Both matrices are 3x2, which means they are compatible for subtraction.

Step 2: Subtract Corresponding Elements

Subtract each element in the first matrix from the corresponding element in the second matrix. Calculate as follows:\[\begin{bmatrix} 1 & -4 \ 2 & -3 \ -8 & 4 \end{bmatrix}-\begin{bmatrix} -6 & 9 \ -2 & 5 \ -7 & -12 \end{bmatrix} = \begin{bmatrix} 1 - (-6) & -4 - 9 \ 2 - (-2) & -3 - 5 \ -8 - (-7) & 4 - (-12) \end{bmatrix}\]

Step 3: Calculate the Resulting Elements

Complete the element-wise subtraction:- For the first row: \[ 1 - (-6) = 1 + 6 = 7 \] and \[ -4 - 9 = -13 \].- For the second row: \[ 2 - (-2) = 2 + 2 = 4 \] and \[ -3 - 5 = -8 \].- For the third row: \[ -8 - (-7) = -8 + 7 = -1 \] and \[ 4 - (-12) = 4 + 12 = 16 \].Thus, the resulting matrix is: \[\begin{bmatrix} 7 & -13 \ 4 & -8 \ -1 & 16 \end{bmatrix} \]

Step 4: Write Down the Final Answer

Write the final resulting matrix from the operation:\[\begin{bmatrix} 7 & -13 \ 4 & -8 \ -1 & 16 \end{bmatrix} \].

Key Concepts

Matrix Operations

Matrix operations involve performing mathematical calculations on matrices, which are rectangular arrays composed of numbers. They are fundamental in various fields, including computer graphics, engineering, and data science.

Common operations include addition, subtraction, multiplication, and finding inverses or determinants. It's important to understand that each operation has specific rules, particularly regarding the dimensions of the matrices involved. In this exercise, we focus on matrix subtraction, a type of element-wise operation.

Before performing any matrix operation, you must ensure that the operation is defined for the matrices in question. Some operations, such as addition and subtraction, require matrices to have identical dimensions.

When discussing matrix subtraction, specifically, it's a good idea to conceptualize it as taking an element from one matrix and subtracting a corresponding element from another matrix.

Element-wise Subtraction

Element-wise subtraction is a straightforward operation in matrix mathematics where you subtract corresponding elements from two matrices. It requires matrices to be of the same dimension, as each element of the first matrix is directly subtracted by the corresponding element of the second matrix.

Here's how you perform this type of subtraction:

• Identify the matrices that need to be subtracted and confirm they have the same dimensions.
• For each element, perform the subtraction by taking the number from the first matrix and subtracting the number from the corresponding position in the second matrix.

For example, in the given exercise, the element in the first row and first column of the first matrix, which is 1, is subtracted by the element in the first row and first column of the second matrix, which is -6. Using arithmetic rules, subtracting a negative number turns into addition, so the result is 7.

This process is repeated across all elements to yield a new matrix consisting of the results of these element-wise operations.

Matrix Dimensions

Matrix dimensions play a critical role when performing matrix operations like addition and subtraction. A matrix is defined by its number of rows and columns, given in the format \(m \times n\), where \(m\) is the number of rows, and \(n\) is the number of columns.

For element-wise operations such as subtraction, both matrices must have the same dimensions. This is because each element in one matrix is paired with exactly one element in the same position of the other matrix. If the matrices had different dimensions, such an operation wouldn't be possible.

In the exercise, both matrices are of size \(3 \times 2\), meaning they each have 3 rows and 2 columns. This confirms that they are compatible for subtraction.

Always verify the dimensions before starting the operation to avoid any errors, as mismatched dimensions are a common pitfall in matrix mathematics.