Problem 20 - Precalculus with Limits

Question

In Exercises 19-24, evaluate the expression. \(\left[ \begin{array}{r} 6 & 8 \\ -1 & 0 \end{array} \right] + \left[ \begin{array}{r} 0 & 5 \\ -3 & -1 \end{array} \right] + \left[ \begin{array}{r} -11 & -7 \\ 2 & -1 \end{array} \right]\)

Step-by-Step Solution

Answer

Therefore, the sum of the given matrices is \[\left[ \begin{array}{r} -5 & 6 \ -2 & -2 \end{array} \right]\].

Step 1: Identify Corresponding Entries

Each matrix has two rows and two columns. The entry in the first row and first column of each matrix corresponds to the entry in the first row and first column of the other matrices. Similarly for the other entries.

Step 2: Add Corresponding Entries

To add matrices, add together each pair of corresponding entries. For the entry in the first row and first column, \(6 + 0 -11 = -5\). For the entry in the first row and second column, \(8 + 5 -7 = 6\). For the entry in the second row and first column, \(-1 -3 + 2 = -2\). And for the entry in the second row and second column, \(0 -1 -1 = -2\).

Step 3: Write the Result as a Matrix

Combine the entries from step 2 into a single matrix. The entry obtained from the first row and first column entries will be the entry in the first row and first column of the result, and similarly for the other entries. Thus, the result is the matrix \[\left[ \begin{array}{r} -5 & 6 \ -2 & -2 \end{array} \right]\].

Key Concepts

Matrix Operations

Matrix operations are fundamental in mathematics, especially in fields like Linear Algebra and Precalculus. When dealing with matrices, one of the most common operations is addition. In matrix addition, you add the corresponding elements from each matrix. For example, if you have two matrices of the same dimensions, you add each element of the first matrix to the element that occupies the same position in the second matrix.

Here are some key points to remember about matrix addition:

• Matrices must have the same dimensions (e.g., both must be 2x2 matrices) to be added together.
• The result of adding two matrices is a new matrix, with each element being the sum of the corresponding elements from the original matrices.
• This process is repeated for each element in the matrix until all have been added.

The operation is as straightforward as simple arithmetic, ensuring each piece of data is accurately combined to form a new set of information. Practice is key to becoming fluent in these operations.

Linear Algebra

Linear Algebra is a branch of mathematics focusing on linear equations, vectors, and matrices. It forms the foundation of many advanced mathematical, scientific, and engineering concepts. One of its core topics is matrix theory, which includes operations like addition, multiplication, and finding determinants.

In Linear Algebra, understanding matrices and their operations is crucial because:

• Matrices can represent complex systems and processes succinctly.
• They are used to solve systems of linear equations efficiently.
• They provide a way to easily handle and manipulate large amounts of data.

By gaining proficiency in matrix operations, you open doors to more advanced topics such as eigenvalues, eigenvectors, and matrix transformations, which have practical applications in computer graphics, optimizations, and more. Being comfortable with matrices is akin to learning a new language in mathematics.

Precalculus

In Precalculus, matrices are introduced as part of studying functions, systems of equations, and transformations. It sets the stage for higher-level mathematics subjects by offering a sneak peek into the concepts of Linear Algebra.

Matrices in Precalculus are particularly useful because:

• They simplify solving systems of equations, which are common in many precalculus problems.
• Matrices are a stepping stone to understanding more complex transformations and space manipulations in calculus.
• They build foundational skills for analytical thinking and problem-solving in other areas of mathematics.

Learning about matrices in Precalculus gives students the tools to manage and interpret data systematically, preparing them for the analytical challenges of Calculus and beyond.