Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. \(\begin{array}{ll}\text { Original Matrix } & \text { New Row-Equivalent Matrix }\end{array}\) $$ \left[\begin{array}{rrr} -2 & 5 & 1 \\ 3 & -1 & -8 \end{array}\right] \quad\left[\begin{array}{rrr} 13 & 0 & -39 \\ 3 & -1 & -8 \end{array}\right] $$

Step-by-Step Solution

Answer

The identified row operation is: R1 = -13/2 * R1 - 5 * R2

Step 1: Identify the changes in the first row

In the original matrix, the first row is (-2, 5, 1), but in the row-equivalent matrix, the first row is (13, 0, -39). From this, it can be noted that the first and third numbers have been multiplied by -13/2, but the second number has been reduced to zero.

Step 2: Identify the operations applied

The first element in the row, -2, is multiplied by -13/2 to get 13. The third element, 1, is also multiplied by -13/2 to get -39. The second element, 5, is reduced to zero by subtracting 5 times the second row's second element (-1). Therefore, the operations applied are multiplication of the first and third elements by -13/2, and subtraction of 5 times the second row from the first row.

Step 3: Summarization

The row operation being performed to get to the new row equivalent matrix is: Multiply the first row by -13/2 and subtract 5 times the second row from the first.

Key Concepts

Row-Equivalent Matrices

Row-equivalent matrices are an important concept in linear algebra. They represent matrices that can be transformed into one another through a series of elementary row operations. This means that, although the matrices might look different, they actually convey the same system of equations or the same linear transformation when represented in a different format.

Elementary row operations include:

• Swapping two rows
• Multiplying a row by a non-zero constant
• Adding a multiple of one row to another row

Understanding these operations is essential. They allow us to simplify matrices, solve systems of linear equations, and find their row echelon or reduced row echelon forms. When you see a change between an original matrix and a row-equivalent matrix, you can trace the operations back step-by-step, as demonstrated in the exercise you've encountered. In the provided example, the first row was manipulated to achieve the transformation, showing the practical application of these operations.

Matrix Multiplication

Matrix multiplication is a fundamental operation in the world of linear algebra that combines two matrices to produce another matrix. It is essential to understand matrix multiplication, which is not simply element-wise multiplication. Instead, it involves a process where each element of the resulting matrix is calculated by taking the dot product of the corresponding row of the first matrix and the column of the second matrix.

For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. If you multiply a matrix, say A with dimensions m x n, by matrix B with dimensions n x p, the resulting matrix will have dimensions m x p.

• Remember that matrix multiplication is not commutative: AB ≠ BA.
• Utilize the distributive and associative properties during calculations.

Knowing how to multiply matrices allows for deeper insights into transformations and systems of equations, crucial for various applications within linear algebra.

Linear Algebra

Linear algebra is the branch of mathematics that focuses on linear equations, matrices, vector spaces, and calculus of linear functions. It is a powerful tool used in various fields including engineering, physics, economics, and more. Linear algebra provides a framework for understanding complex systems, modeling real-world phenomena, and solving problems efficiently.

Key components of linear algebra include:

• Matrices and determinants, which are used to handle data and perform transformations.
• Vector spaces and vector operations, serving as the foundation for linear systems.
• Linear transformations, illustrating mappings between different vector spaces.

By mastering the essentials of linear algebra, you gain the ability to handle large-scale data analysis and solve multidimensional problems, enhancing both mathematical knowledge and practical application skills.