Problem 18 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The result of the matrix addition is \(\begin{bmatrix} 6 & 6 \\ -12 & 9 \end{bmatrix}\).

Step 1: Check Matrix Dimensions

Before proceeding with the matrix addition, we need to verify that both matrices have the same dimensions. The first matrix is a 2x2 matrix and the second matrix is also a 2x2 matrix. Since both matrices have the same dimensions, they can be added together.

Step 2: Perform Matrix Addition

To add two matrices, add their corresponding elements. Let's perform the addition element by element:- First row, first column: \(9 + (-3) = 6\)- First row, second column: \(4 + 2 = 6\)- Second row, first column: \(-8 + (-4) = -12\)- Second row, second column: \(2 + 7 = 9\)This results in the matrix:\[\begin{bmatrix} 6 & 6 \ -12 & 9 \end{bmatrix}\]

Step 3: Verify Solution

Check the calculated matrix to ensure all operations were conducted correctly. Re-evaluate the addition for each element if necessary to confirm accuracy.

Key Concepts

Understanding Matrices

Matrices are a fundamental concept in mathematics and are used in a variety of applications in science and engineering. A matrix is essentially a rectangular array of numbers, symbols, or expressions arranged in rows and columns.

In the context of matrix addition, understanding the structure of matrices is crucial. Each number in a matrix is called an element, and matrices are typically denoted by capital letters such as \( A \) or \( B \). When matrices are written, they appear in brackets or parentheses, clearly displaying the rows and columns.

• Each horizontal line of elements is a row.
• Each vertical line of elements is a column.

To perform operations such as addition, the arrangement of the elements must be compatible. This means that each matrix involved must have the same number of rows and the same number of columns. Only then can we combine their respective elements.

Matrix Dimensions in Operations

The dimensions of a matrix are a critical factor when it comes to matrix operations. Matrix dimensions are given in the format "\( m \times n \)," where \( m \) is the number of rows and \( n \) is the number of columns. In the original exercise, both matrices are \( 2 \times 2 \), meaning that each has 2 rows and 2 columns.

For matrix addition (or subtraction) to be performed, the matrices must have identical dimensions. This requirement ensures that each element in a matrix corresponds to exactly one element in the same position in the other matrix.

• If the dimensions of the two matrices match, we can proceed with addition.
• If they don't, matrix addition is not possible.

Knowing the dimensions is like checking the compatibility of a lock and key. Only when they fit can you "unlock" the ability to solve a matrix operation like addition.

Element-wise Operations in Matrix Addition

Element-wise operations are at the heart of matrix addition. In such operations, we simply add the corresponding elements from each matrix together. This means, for each position in the first matrix, you have a matching position in the second matrix that you're adding to.

Think of element-wise operations like aligning two grids of numbers and summing each overlapping pair of numbers. It's done row by row and column by column, and here's an example from the exercise:

• Element at first row, first column: Add \( 9 \) from first matrix to \( -3 \) from second, yielding: \( 9 + (-3) = 6 \).
• Element at first row, second column: Add \( 4 \) to \( 2 \), resulting in \( 4 + 2 = 6 \).
• Follow similar steps for subsequent elements.

This creates a new matrix, termed the resulting matrix, composed of these calculated sums for each corresponding position. It’s a straightforward process once you grasp the alignment of elements. You may also want to verify each calculation to avoid errors in the resulting matrix.