Problem 12 - College Algebra with Modeling and Visualization

Question

Is the matrix in row-echelon form? (a) \(\left[\begin{array}{rr|r}1 & 3 & 2 \\ 0 & -1 & -1\end{array}\right]\) (b) \(\left[\begin{array}{rrr|r}1 & 3 & -1 & 8 \\ 0 & 1 & 5 & 3 \\ 0 & 0 & 0 & 0\end{array}\right]\) (c) \(\left[\begin{array}{rrr|r}0 & 0 & 1 & 1 \\ 0 & 1 & 7 & 9 \\ 1 & 2 & -1 & 11\end{array}\right]\)

Step-by-Step Solution

Answer

Matrix (b) is in row-echelon form.

Step 1: Understanding Row Echelon Form

A matrix is in row-echelon form if it satisfies the following conditions: 1) All nonzero rows are above any rows of all zeros; 2) The leading entry of each nonzero row after the first occurs to the right of the leading entry of the previous row; 3) The leading entry in any nonzero row is 1 (also called a leading 1); 4) All entries in a column below a leading 1 are zero.

Step 2: Check Matrix (a)

The matrix \( \left[\begin{array}{rr|r}1 & 3 & 2 \ 0 & -1 & -1\end{array}\right] \) has the first leading entry as 1, and it is arranged such that all nonzero rows are above any zero rows, and each leading entry is to the right of the leading entry in the row above. However, the second row's leading entry is not positive. Therefore, it is not in row-echelon form as it fails condition 3.

Step 3: Check Matrix (b)

For the matrix \( \left[\begin{array}{rrr|r}1 & 3 & -1 & 8 \ 0 & 1 & 5 & 3 \ 0 & 0 & 0 & 0\end{array}\right] \): The matrix has the leading entry of the first non-zero row as 1, the second row has a leading 1, which is to the right of the first row's leading 1, and the last row is all zeros. This satisfies all the conditions for row-echelon form.

Step 4: Check Matrix (c)

The matrix \( \left[\begin{array}{rrr|r}0 & 0 & 1 & 1 \ 0 & 1 & 7 & 9 \ 1 & 2 & -1 & 11\end{array}\right] \): The leading entry is in the first column, which is not allowed as it should be in each subsequent column to the right for each row. This matrix does not satisfy the row-echelon form requirements.

Key Concepts

Matrix Algebra

Matrix algebra is an essential part of linear algebra that involves various operations on matrices, such as addition, subtraction, multiplication, and finding determinants and inverses. Matrices are rectangular arrays of numbers organized in rows and columns. They are often used to represent complex systems of linear equations and transformations.

One of the fundamental concepts of matrix algebra is the ability to manipulate rows and columns to achieve a particular form, such as the row-echelon form. This form simplifies the process of solving linear equations.

Matrix operations are governed by specific rules:

• Addition and subtraction require matrices to have the same dimensions.
• Multiplication involves the dot product of rows and columns from two matrices, which must be conformable.
• The transpose of a matrix involves flipping rows and columns.

Understanding these basic matrix operations is crucial when analyzing and solving systems of equations using matrices.

Linear Equations

Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable. They are fundamental in mathematics as they represent straight lines when graphed in two-dimensional space.

In the context of systems of linear equations, we often encounter multiple equations that need to be solved simultaneously. This system is typically represented in matrix form, where the coefficients of the variables form the matrix, and the constants are positioned in an additional column, creating an augmented matrix.

Using matrices to solve linear equations is efficient, as it allows for streamlined operations like Gaussian elimination. This process transforms the original matrix into simpler forms: typically row-echelon form or reduced row-echelon form. This makes it easier to interpret solutions, especially in terms of finding unique solutions, infinitely many solutions, or confirming if no solution exists.

• Unique solutions arise when there is precisely one solution.
• Infinite solutions occur when multiple equations represent the same line or plane.
• No solution is present when equations form parallel lines or planes that never intersect.

Leading Entry

The concept of a leading entry is crucial when working with matrices in row-echelon form. This entry is the first non-zero number in a row of a matrix and plays a key role in organizing matrices to simplify solving systems of linear equations.

In a row-echelon form, each leading entry in a non-zero row must be 1, known as a leading 1, and appears to the right of any leading entries in the row above it. Additionally, all entries below a leading entry within its column must be zero, ensuring a "staircase" pattern down the matrix.

For example, in a matrix with rows like \(egin{bmatrix}1 & 3 & -1 & 8 \ 0 & 1 & 5 & 3 \ 0 & 0 & 0 & 0egin{bmatrix}\), each row starts with a 1, and the positions of these leading 1s shift progressively to the right as you move down the rows.
Identifying and manipulating leading entries helps in mastering the techniques of matrix row reductions and understanding the solution sets of linear equations efficiently.