Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the matrix in row-echelon form. Remember that the row-echelon form of a matrix is not unique. $$\left[\begin{array}{ccc} 1 & -3 & 2 \\ 5 & 0 & 7 \end{array}\right]$$

Step-by-Step Solution

Answer

The row-echelon form of the given matrix is: \[\left[\begin{array}{ccc} 1 & -3 & 2 \ 0 & 1 & -1/5 \end{array}\right]\]

Step 1: Identify the pivot of the first row

The pivot of the first row in this scenario is 1, which is the first number in the first row and is not zero.

Step 2: Make the pivot of the second row zero

To do this, multiply the first row by -5 and add it to the second row. Calculation: -5(1 -3 2) + (5 0 7) = (0 15 -3). Now the matrix looks like this: \[\left[\begin{array}{ccc} 1 & -3 & 2 \ 0 & 15 & -3 \end{array}\right]\]

Step 3: Make the pivot of the second row 1

Divide the second row by 15. The final matrix in row-echelon form is: \[\left[\begin{array}{ccc} 1 & -3 & 2 \ 0 & 1 & -1/5 \end{array}\right]\]

Key Concepts

Matrix Algebra

Matrix algebra is an essential component in the field of mathematics, particularly in the study of linear equations and transformations. Essentially, matrices are rectangular arrays of numbers arranged in rows and columns that can be used to represent systems of linear equations or to perform linear transformations on vectors.

Operations such as addition, subtraction, and multiplication follow specific rules that preserve the properties of matrices. For example, two matrices can only be added or subtracted if they have the same dimensions. Multiplication, however, is more complex; the number of columns in the first matrix must equal the number of rows in the second matrix for the operation to be possible.

In our exercise, transforming the given matrix into row-echelon form is a part of matrix algebra. It involves using various operations to simplify matrices into a form that makes it easier to solve linear equation systems or to understand the structure of the matrix. The row-echelon form is a step towards simplifying the matrix, even though it is not unique, meaning there can be more than one correct row-echelon form for the same initial matrix.

Pivot Element

The pivot element in a matrix is the first non-zero element of a row, read from left to right. In the context of matrix algebra, it holds significant importance, particularly when transforming a matrix to its row-echelon form. The pivot element is typically chosen to be equal to 1, and it is used to create zeros below it, which helps clear out the column it resides in for easier manipulation of the system.

During the transformation process, one of the goals is to achieve a leading 1 in each row, if possible. This '1' becomes the pivot element for that particular row. In our exercise, the pivot element for the first row is already 1, which is ideal. The pivot elements are key in determining the steps needed to achieve row-echelon form and ultimately solving the system of equations that the matrix represents.

Elementary Row Operations

Elementary row operations are the tools we use to manipulate matrices in order to find solutions to systems of linear equations or to transform the matrix into a simpler form, such as row-echelon form. There are three types of elementary row operations:

• Row switching, where two rows are swapped.
• Row multiplication, where a row is multiplied by a non-zero constant.
• Row addition, where a multiple of one row is added to another to change that row.

These operations are fundamental to matrix algebra because they maintain the equivalency of the system that the matrix represents.

In the given exercise, row multiplication and addition were used to transform the matrix. In step 2, we multiplied the first row by -5 and added it to the second row to create a zero below the first pivot element. This combination of multiplication and addition is a powerful technique used to simplify matrices without losing their inherent properties.