Question

State which of the following matrices are in echelon form: (a) \(\left(\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 1\end{array}\right)\) (b) \(\left(\begin{array}{llll}1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{array}\right)\) (c) \(\left(\begin{array}{ccccc}1 & 2 & 3 & 1 & 4 \\ 0 & 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 & 2\end{array}\right)\) (d) \(\left(\begin{array}{cccc}1 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 4 & 3\end{array}\right)\) (e) \(\left(\begin{array}{cccccc}0 & 0 & 1 & 2 & 0 & -4 \\ 0 & 0 & 0 & 0 & 1 & 3\end{array}\right)\)

Short answer

(a) \(\left(\begin{array}{lll}1 & 0 & 1 \\\ 0 & 1 & 1\end{array}\right)\) (b) \(\left(\begin{array}{llll}1 & 1 & 1 & 0 \\\ 0 & 0 & 0 & 1 \\\ 0 & 0 & 0 & 0\end{array}\right)\) (c) \(\left(\begin{array}{ccccc}1 & 2 & 3 & 1 & 4 \\\ 0 & 0 & 1 & -1 & 0 \\\ 0 & 0 & 0 & 1 & 2\end{array}\right)\) (d) \(\left(\begin{array}{cccc}1 & 0 & 1 & -3 \\\ 0 & 0 & 0 & 0 \\\ 0 & 1 & 4 & 3\end{array}\right)\) (e) \(\left(\begin{array}{cccccc}0 & 0 & 1 & 2 & 0 & -4 \\\ 0 & 0 & 0 & 0 & 1 & 3\end{array}\right)\) Answer: The matrices in echelon form are (a), (c), and (e).

Step-by-step solution

Matrix (a)

For the given matrix (a): \(\left(\begin{array}{lll}1 & 0 & 1 \\\ 0 & 1 & 1\end{array}\right)\). This matrix fulfills all the echelon form conditions. All nonzero rows are above rows of all zeros (there is no row with all zeros), the leading coefficient in the second row (1) is strictly to the right of the pivot of the row above it (also 1), and all entries in a column below the pivot are zeros. Therefore, matrix (a) is in echelon form.

Matrix (b)

For the given matrix (b): \(\left(\begin{array}{llll}1 & 1 & 1 & 0 \\\ 0 & 0 & 0 & 1 \\\ 0 & 0 & 0 & 0\end{array}\right)\). This matrix fulfills the condition that all nonzero rows are above any rows of all zeros, and the leading coefficient in the second row (1) is strictly to the right of the pivot of the row above it (also 1). However, it does not fulfill the condition that all entries in a column below the pivot should be zeros, since there is a 1 in the second column of the first row. Therefore, matrix (b) is not in echelon form.

Matrix (c)

For the given matrix (c): \(\left(\begin{array}{ccccc}1 & 2 & 3 & 1 & 4 \\\ 0 & 0 & 1 & -1 & 0 \\\ 0 & 0 & 0 & 1 & 2\end{array}\right)\). This matrix fulfills all echelon form conditions. All nonzero rows are above rows of all zeros (there is no row with all zeros), the leading coefficient in the second row (1) is strictly to the right of the pivot of the row above it (also 1), and all entries in a column below the pivot are zeros. Therefore, matrix (c) is in echelon form.

Matrix (d)

For the given matrix (d): \(\left(\begin{array}{cccc}1 & 0 & 1 & -3 \\\ 0 & 0 & 0 & 0 \\\ 0 & 1 & 4 & 3\end{array}\right)\). This matrix does not fulfill the condition that the leading coefficient of a nonzero row is always strictly to the right of the pivot of the row above it, because the pivot in the third row (1) is in the second column, which is not strictly to the right of the pivot of the row above it (the first column). Therefore, matrix (d) is not in echelon form.

Matrix (e)

For the given matrix (e): \(\left(\begin{array}{cccccc}0 & 0 & 1 & 2 & 0 & -4 \\\ 0 & 0 & 0 & 0 & 1 & 3\end{array}\right)\). This matrix fulfills all echelon form conditions. All nonzero rows are above rows of all zeros (there is no row with all zeros), the leading coefficient in the second row (1) is strictly to the right of the pivot of the row above it (also 1), and all entries in a column below the pivot are zeros. Therefore, matrix (e) is in echelon form.

Key concepts

Matrix Algebra

Matrix algebra is a crucial branch of mathematics that deals with the study and manipulation of matrices. In the context of echelon forms, matrix algebra becomes particularly important. An echelon form of a matrix is a kind of step-stone to the solution of systems of linear equations, which are represented by the matrix.

Echelon form, specifically, has certain characteristics: the so-called leading entries (or pivots) of nonzero rows appear to the right of the leading entry of the row above, all entries below a pivot must be zero, and any zero rows are positioned at the bottom of the matrix. Special cases of echelon form include row-echelon form (REF) and reduced row-echelon form (RREF), with additional stipulations on the pivots and the entries above them.

Understanding the concepts of matrix algebra, such as echelon forms, is pillars for more advanced sections of mathematics and applications in various fields, including engineering, computer science, economics, and more. Simplifying matrices to echelon form is a standard method for solving linear systems, which is why this knowledge is fundamental to students in these areas.

Mathematics for Engineers

For engineers, mathematics serves as a language and a toolkit for designing, analyzing, and solving real-world problems. When examining structures, engineers use matrix algebra to handle complex calculations that involve multiple variables and equations with ease. The row-echelon form is particularly useful in engineering as it simplifies matrices from systems of linear equations linked to various engineering problems, such as circuit analysis or statics.

For example, when an engineer needs to determine the forces in a truss system or the voltages in an electrical circuit, they can set up a system of linear equations to model the problem. By converting the system's matrix to an echelon form, the engineer can then proceed to use back-substitution to find the solution more straightforwardly, as each equation becomes simpler to handle going from the bottom row upward.

Therefore, learning how to recognize and manipulate matrices into their echelon forms is an essential skill for engineers, not only for its theoretical importance but for its practical applications that make computational work in engineering more efficient and precise.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It's a foundational topic not just in mathematics, but in numerous applications across science and engineering. Echelon form matrices are one of the many tools linear algebra provides.

In solving systems of linear equations—which is a central problem in linear algebra—row reduction (also known as Gaussian elimination) is employed to transform matrices into echelon form. This process simplifies the set of equations represented by the matrix, making it easier to understand the structure of the system and find solutions. In echelon form, the zeros below pivots reduce the need to consider certain variables when solving a particular row, thereby breaking down complex systems into simpler, manageable parts.

The exercise provided focuses on identifying whether given matrices are in echelon form, which reiterates the importance of recognizing structured matrices as part of the problem-solving process in linear algebra. Mastery of these concepts in linear algebra is pivotal, as it forms the groundwork for further studies and applications in mathematics, physics, computer algorithms, and data analysis.