Problem 53 - Precalculus with Limits

Question

In Exercises 53-56, evaluate the expression. Use the matrix capabilities of a graphing utility to verify your answer. \(\left[\begin{array}{r} 3 & 1 \\ 0 & -2 \end{array}\right]\left[\begin{array}{r} 1 && 0 \\ -2 && 2 \end{array}\right]\left[\begin{array}{r} 1 && 0 \\ 2 && 4 \end{array}\right]\)

Step-by-Step Solution

Answer

The matrix obtained after carrying out all the multiplication operations is \( \left[ \begin{array}{r} -1 & 8 \\ -4 & -8 \end{array} \right] \)

Step 1: Multiply First and Second Matrices

Begin the operation by multiplying the two first matrices element by element. The product of two matrices is determined by row by column multiplication and combination. The first element in the new matrix is obtained by multiplying each element in the first row of the first matrix by the corresponding element in the first column of the second matrix and summing these values: \(3*1 + 1*(-2) = 1\). This process is repeated for the remaining elements of the new matrix.

Step 2: Multiply the Resulting Matrix with the Third Matrix

The matrix resulting from step 1 is then multiplied by the third matrix in the same manner. Again, each element in the new matrix is calculated by multiplying corresponding elements in the row of the first matrix and the column of the second matrix followed by summation.

Step 3: Present the final Result

The final result should be represented as a 2x2 Matrix.

Key Concepts

Matrix Operations

Matrix operations involve a variety of arithmetic operations between matrices. Multiplication is one of the key operations. To multiply two matrices, we use a method called row-by-column multiplication. This means the elements in the rows of the first matrix get multiplied by the corresponding elements in the columns of the second matrix. Then, the products are added up to form a single element in the resulting matrix.
For instance, in step 1 of our solution, we multiply the first row of the first matrix \( \begin{pmatrix} 3 & 1 \end{pmatrix} \) by the first column of the second matrix \( \begin{pmatrix} 1 \ -2 \end{pmatrix} \), and the resulting sum defines an entry in the product matrix. This procedure is repeated for each element in the matrices until the new matrix is fully populated.

• Matrix size must be compatible; the number of columns in the first matrix must equal the number of rows in the second matrix.
• Order matters, as switching the matrices can yield different results.

Matrix multiplication is fundamental in various applications, including computer graphics, solving systems of equations, and more.

Matrices

Matrices are mathematical structures composed of rows and columns. They are used to represent and solve linear equations, transform geometric figures, perform data operations, and much more.
A matrix is generally denoted as \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), where each letter represents a different element. The dimensions of a matrix are given by "rows x columns."

• A 2x2 matrix has 2 rows and 2 columns.
• Matrices can be square, rectangular, or other shapes based on their dimensions.

Elements within the matrix can be numbers, expressions, or even other mathematical entities. Understanding matrices is crucial because they provide a framework to manipulate complex data and perform various matrix operations.

Mathematical Expression Evaluation

Evaluating a mathematical expression involves calculating the result by following specific mathematical rules and operations. In the context of matrices, it often involves performing step-by-step operations.
When evaluating matrix expressions, it is essential to adhere to the order of operations, specifically:
1. Begin from the leftmost operation, especially if parentheses prioritize a certain part.
2. Perform matrix multiplications one step at a time.
3. Ensure each resulting matrix is well-formed before proceeding to further calculations.

• Check each result for errors before moving on to the next step.
• Use graphing utilities or matrix calculators to verify complex operations.

Evaluating matrix expressions efficiently requires an understanding of both the basics of matrices and the order of operations, ensuring accuracy and correctness in calculations.