Problem 12 - Algebra and Trigonometry

Question

7–14 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. $$\left[\begin{array}{llll}{1} & {0} & {0} & {1} \\ {0} & {1} & {0} & {2} \\\ {0} & {0} & {1} & {3}\end{array}\right]$$

Step-by-Step Solution

Answer

(a) Yes; (b) Yes; (c) \(x = 1\), \(y = 2\), \(z = 3\).

Step 1: Understanding Row-Echelon Form

A matrix is in row-echelon form if: 1) All non-zero rows are above any rows of all zeroes, 2) The leading entry of each non-zero row after the first occurs to the right of the leading entry of the previous row, 3) The leading entry in any non-zero row is 1, and 4) Entries in columns below a leading 1 are zero.

Step 2: Checking Row-Echelon Form

The given matrix is \(\begin{bmatrix} 1 & 0 & 0 & 1 \ 0 & 1 & 0 & 2 \ 0 & 0 & 1 & 3 \end{bmatrix}\). Here, all leading entries in non-zero rows are 1, and each leading 1 in a row is to the right of the leading 1 in the row above it. There are no rows of all zeroes. Hence, this matrix is in row-echelon form.

Step 3: Understanding Reduced Row-Echelon Form

A matrix is in reduced row-echelon form if: 1) It is in row-echelon form, 2) The leading entry in each non-zero row is 1, and is the only non-zero entry in its column.

Step 4: Checking Reduced Row-Echelon Form

In the matrix \(\begin{bmatrix} 1 & 0 & 0 & 1 \ 0 & 1 & 0 & 2 \ 0 & 0 & 1 & 3 \end{bmatrix}\), each leading 1 is the only non-zero entry in its column, which satisfies the conditions for being in reduced row-echelon form. Thus, the matrix is in reduced row-echelon form.

Step 5: Identifying the System of Equations

The augmented matrix represents the system of equations by considering each row as an equation. For the given matrix, the first row corresponds to \(x = 1\), the second row corresponds to \(y = 2\), and the third row corresponds to \(z = 3\). Thus, the system of equations is: \(x = 1\), \(y = 2\), \(z = 3\).

Key Concepts

Reduced Row-Echelon Form

Matrices can be quite confusing, especially when it comes to understanding the different forms such as reduced row-echelon form. At its core, a matrix is in reduced row-echelon form if:

• It is first in row-echelon form.
• The leading entry in each non-zero row is 1.
• This leading 1 is the only non-zero entry in its column.

In simpler terms, all columns containing leading 1s look like the columns of the identity matrix.
The given matrix \(\begin{bmatrix} 1 & 0 & 0 & 1 \ 0 & 1 & 0 & 2 \ 0 & 0 & 1 & 3 \end{bmatrix}\) satisfies all these conditions. Each leading 1 is isolated in its column, which makes this matrix not only in row-echelon form but also in reduced row-echelon form.
Understanding this definition is crucial for solving systems of equations effectively and simplifying matrix-related computations.

Augmented Matrix

An augmented matrix is a great tool in linear algebra, as it combines coefficients of variables with the constants from equations in a neat tabular form. Visualizing a system of equations this way helps in applying various matrix techniques to solve it.
In our case, the matrix given is:\[\begin{bmatrix} 1 & 0 & 0 & 1 \ 0 & 1 & 0 & 2 \ 0 & 0 & 1 & 3 \end{bmatrix}\]The last column here acts as the constants from a system of equations, while the preceding columns represent the coefficients of variables (e.g., x, y, z).
Interpreting each row, we transform the matrix back into equations that it originally represented. This is a powerful way of understanding relations among different variables in a system and manipulating those using matrix operations.

System of Equations

When we talk about a system of equations, we're discussing a collection of equations that share two or more unknowns. These equations are solved simultaneously to find big-picture solutions for these unknowns.
For the matrix presented:\[\begin{bmatrix} 1 & 0 & 0 & 1 \ 0 & 1 & 0 & 2 \ 0 & 0 & 1 & 3 \end{bmatrix}\]we derive the system of equations from its rows:

• The first row translates to \(x = 1\).
• The second row becomes \(y = 2\).
• The third row gives \(z = 3\).

Here, each equation is simple and shows that the unknowns are explicitly solved, indicating a consistent and independent system. Systems of equations often involve more complex relationships, but representing them in matrices and using matrix algebra can streamline the process of finding solutions.