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. $$\left[\begin{array}{llll} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \end{array}\right]$$

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.
$$\left[\begin{array}{llll}
1 & 0 & 0 & 1 \\
0 & 1 & 0 & 2 \\
0 & 0 & 1 & 3
\end{array}\right]$$

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations that share common variables. The key goal is to find the values of these variables that satisfy all the equations simultaneously. When such systems are expressed in matrix form through an augmented matrix, methods like Gaussian elimination can be used to find a unique solution, infinitely many solutions, or determine that no solution exists.

Augmented Matrix

An augmented matrix is a compact representation of a system of linear equations where the coefficients and the constants are included. The matrix is arranged such that each row corresponds to an equation of the system, with the columns representing coefficients for the variables followed by the constant term. This format is especially useful in applying row operations to solve the system.

Reduced Row Echelon Form

A matrix goes further than row echelon form and is considered in reduced row echelon form when, in addition to satisfying the conditions of row echelon form, every leading entry is 1 and is the only nonzero entry in its column. This stronger form often simplifies both the solving of linear systems and the demonstration of uniqueness of solutions in the context of linear algebra.

Row Echelon Form

A matrix is in row echelon form if all nonzero rows are above any rows of all zeros, each leading entry in a nonzero row is strictly to the right of the leading entry in the row above it, and all entries in a column below a leading entry are zeros. This concept is central in processes like Gaussian elimination where the structure of the row echelon form simplifies the process of solving linear systems.

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