Problem 27 - Algebra and Trigonometry Real Mathematics, Real People

Question

Fill in the blank(s) using elementary row operations to form a row-equivalent matrix. $$\begin{aligned} &\left[\begin{array}{ccc} 1 & 4 & 3 \\ 2 & 10 & 5 \end{array}\right]\\\ &\left[\begin{array}{rrr} 1 & 4 & 3 \\ 0 & & -1 \end{array}\right] \end{aligned}$$

Step-by-Step Solution

Answer

The completed second matrix is \[\begin{array}{rrr} 1 & 4 & 3 \ 0 & 2 & -1 \end{array}\]\.

Step 1: Identify operation used

Observe the two matrices, the first row in both matrices are identical. The elementary row operation has been performed on the second row of the first matrix. The 2 in the second row first column of the first matrix has been converted to 0 in the second matrix, suggesting the operation is subtraction of 2 times the first row from the second row.

Step 2: Perform operation to find missing value

Apply the identified operation (subtracting two times the first row from the second row) to find the missing value. That is the second value of the changed row computed as: \(10 - 2 \times 4 = 2\).

Step 3: Insert missing value into the matrix

Place the computed value in the second row second column of the second matrix.

Key Concepts

Row-Equivalent Matrix

A row-equivalent matrix is a transformed version of an original matrix, achieved through elementary row operations. These operations are vital in linear algebra to simplify matrices while maintaining their essential properties. Row-equivalent matrices display the same row span (the set of all possible linear combinations of their rows) as the original.

This means that two row-equivalent matrices represent the same linear system of equations. They provide the same solutions. Achieving row equivalence helps in determining the rank of a matrix, solving systems of linear equations, and simplifying matrices for further computation.

• **Equivalence through Transformation:** By applying a series of elementary row operations - such as row swapping, row scaling, or row addition/subtraction - we transform an original matrix into its row-equivalent form.
• **Properties Preserved:** These operations do not alter the linear independence or dependence of the rows, maintaining the solutions to systems of equations represented by the matrix.
• **Usage:** Primarily, row-equivalence is used to achieve reduced row-echelon form, aiding in clear solutions to complex systems.

Matrix Operations

Matrix operations include a variety of procedures, such as addition, subtraction, and multiplication, that allow mathematicians to manipulate matrices and solve equations. Elementary row operations, a subset of these operations, are used extensively to transform matrices. This manipulation helps in solving linear systems and finding matrix inverses when applicable.

Understanding these operations is crucial, as it supports further advancements in studies of linear algebra, differential equations, and other mathematical fields.

• **Addition/Subtraction:** For matrices of the same dimension, these operations are performed element-wise, allowing them to be combined or adjusted similarly to ordinary arithmetic operations.
• **Multiplication:** This involves dot products and is fundamental for matrix equations. It is not commutative like normal multiplication, meaning the order in which matrices are multiplied matters.
• **Elementary Operations:** These include specific transformations like row swapping, row multiplication (by a non-zero scalar), or adding a multiple of one row to another. These are designed to help simplify matrices and reach solutions efficiently.

Algebraic Computation

Algebraic computation refers to the process of performing calculations using the principles of algebra. In the context of matrices, this involves using row operations and understanding how different transformations affect matrix properties. Elemental in solving systems of linear equations, these calculations often streamline complex processes.

In the exercise given, algebraic computation involves determining how transformations alter the matrix, ensuring the correct application of operations, and calculating any unknown values, as demonstrated by finding the missing value in the matrix using the identified operation: subtracting doubles of one row from another.

• **Precision in Calculation:** Every step in algebraic computation is precise and intentional, requiring accuracy to maintain matrix integrity.
• **Application:** Used extensively in various fields, from engineering to data science, to solve linear equations and perform data transformations necessary for problem-solving.
• **Mathematical Tools:** Leveraging concepts like linear combinations, determinants, and eigenvalues, algebraic computation aids in drawing deeper insights from data represented in matrix form.