Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $$\left[\begin{array}{lllr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The given matrix is in row-echelon form but not in reduced row-echelon form.

Step 1: Check for Row-Echelon Form

Look at the given matrix: \[ \left[\begin{array}{lllr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 2 \end{array}\right] \]. All rows with all zeroes are already at the bottom (there aren't any). The first non-zero element in each row (the pivot) is to the right of the pivot in the row above it. Therefore, the matrix is in row-echelon form.

Step 2: Check for Reduced Row-Echelon Form

For a matrix to be in reduced row-echelon form, the pivot must be 1 and all others in the same column must be 0. In this matrix, we can see that every pivot (the first element of each row) is 1, and all other elements in the columns of pivots are 0, except for the last row. The pivot element in the last row is 2 which disqualifies the matrix from being in reduced row-echelon form. So, the matrix is not in reduced row-echelon form.

Key Concepts

Reduced Row-Echelon Form

Understanding the reduced row-echelon form (RREF) of a matrix is critical for students to grasp the concept of unique solutions in systems of linear equations. In RREF, the matrix has leading 1s, called pivot elements, in each row, and these pivots are the only non-zero numbers in their respective columns. Importantly, the pivot elements appear from top to bottom, left to right, and each pivot is further to the right than the previous one.

When checking for RREF in the exercise, students must ensure that: every leading coefficient (pivot) is 1, each pivot is the only nonzero number in its column, and the pivot columns create an identity matrix within the original matrix. If any of these conditions are not met, as in the case of the provided exercise where the pivot element in the last row was 2 instead of 1, the matrix does not qualify as being in RREF. Although reaching RREF might involve several steps and applying matrix operations, its importance is evident since it simplifies solving linear equations and provides clear insights into the structure of the solution set.

Pivot Elements

Pivot elements serve as a cornerstone for understanding the structure of a matrix. Pivots are the first non-zero elements in each row of a matrix and play a vital role when using Gaussian elimination to solve systems of equations. After transforming a matrix into row-echelon form, pivot elements are identified to assess whether it can be further simplified into reduced row-echelon form.

In the context of the exercise, students should recognize that the presence of pivot elements helps to verify whether the necessary conditions for RREF are satisfied. However, merely having pivot elements is not enough for RREF; their specific values and positions within the matrix affect its categorization. For instance, the exercise illustrates a scenario where the matrix failed to meet RREF criteria due to the last pivot element being 2. Teaching students to identify and properly work with pivot elements is imperative for their success in linear algebra.

Matrix Operations

Matrix operations are essential in transforming matrices into their row-echelon or reduced row-echelon forms. These operations include row addition, row multiplication by non-zero scalars, and row switching. Each of these manipulations is done with the goal of simplifying the matrix to a form where solving linear equations becomes more manageable.

When addressing an exercise like the one given, it's important for students to learn not just to check the form of a matrix but also to understand how to manipulate the matrix to potentially reach RREF. For instance, as part of improving the matrix presented in the exercise, one could divide the last row by 2 to make the pivot element 1 in adherence with RREF. However, the fact that there's a non-zero element in the pivot's column would still prevent it from being in RREF. Furthermore, understanding matrix operations underpins more advanced concepts in linear algebra such as determinant calculation, matrix inversion, and eigenvector computation.