A matrix is given. a. Determine whether the matrix is in...

A matrix is given. a. Determine whether the matrix is in row-echelon form. b. Determine whether the matrix is in reduced row-echelon form. c. Write the system of equations for which the given matrix is the augmented matrix. $$\left[\begin{array}{cccc} 1 & 0 & -7 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$

A matrix is given.
a. Determine whether the matrix is in row-echelon form.
b. Determine whether the matrix is in reduced row-echelon form.
c. Write the system of equations for which the given matrix is the augmented matrix.
$$\left[\begin{array}{cccc}
1 & 0 & -7 & 0 \\
0 & 1 & 3 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]$$

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Key Concepts

Row-Echelon Form

This refers to a matrix organization where each nonzero row begins with a leading entry (which does not necessarily have to be 1), and each leading entry is to the right of the leading entry in the row above it. Additionally, rows consisting entirely of zeros, if any, are positioned at the bottom of the matrix. This hierarchical structure simplifies the process of solving systems of linear equations by making back substitution possible.

Reduced Row-Echelon Form

This is a refined version of the row-echelon form. In addition to the properties of row-echelon form, each leading entry in a nonzero row must be 1, and it must be the only nonzero entry in its column. This form provides a unique (canonical) representation for the corresponding system, making it easier to directly read the solutions to the system of equations.

Augmented Matrix and System of Equations

An augmented matrix is a compact representation of a system of linear equations where the coefficients of the variables form the main part of the matrix, and an additional column represents the constants from the right-hand side. Converting an augmented matrix back to its system of equations involves writing each row as a linear equation, thereby linking matrix methods to classical algebraic techniques.

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