Problem 8 - 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. $$\left[\begin{array}{rrr}{1} & {3} & {-3} \\ {0} & {1} & {5}\end{array}\right]$$

Step-by-Step Solution

Answer

(a) Yes, it's in row-echelon form. (b) No, it's not in reduced row-echelon form. (c) The system is: \(x + 3y = -3\), \(y = 5\).

Step 1: Understand Row-Echelon Form

A matrix is in row-echelon form if: (1) All non-zero rows are above any rows of all zeros. (2) The leading coefficient (first non-zero number from the left) of a non-zero row is always strictly to the right of the leading coefficient of the row above it. (3) The leading coefficient is 1.

Step 2: Examine Given Matrix for Row-Echelon

The given matrix is \(\begin{bmatrix} 1 & 3 & -3 \ 0 & 1 & 5 \end{bmatrix}\). This matrix has two non-zero rows, and the leading coefficient of the second row (1) is to the right of the leading coefficient of the first row. Thus, it satisfies the criteria for row-echelon form.

Step 3: Understand Reduced Row-Echelon Form

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

Step 4: Examine Given Matrix for Reduced Row-Echelon

In the given matrix \(\begin{bmatrix} 1 & 3 & -3 \ 0 & 1 & 5 \end{bmatrix}\), row 1's leading coefficient (1) has other non-zero entries in its column. Specifically, there is a 3 in column 2, meaning the matrix is not in reduced row-echelon form.

Step 5: Translate Matrix to System of Equations

The given matrix represents an augmented matrix of a system of equations where each row corresponds to an equation. The first row, \([1, 3, -3]\), translates to the equation \(x + 3y = -3\), and the second row, \([0, 1, 5]\), represents the equation \(y = 5\).

Key Concepts

Augmented Matrix

An augmented matrix is a very helpful tool when dealing with systems of linear equations. It combines the coefficients of the variables and the constants from their respective equations into one matrix. This format simplifies processes like solving or performing operations on equations. An augmented matrix essentially contains two parts:

• The coefficient matrix: which holds the coefficients of each variable in the equations.
• The augmented column: which contains the constants of each equation that the matrix represents.

For example, if you have a system of equations like:
\[\begin{aligned}& x + 3y = -3 \& y = 5\end{aligned}\]
The augmented matrix would look like:
\[\begin{bmatrix}1 & 3 & | & -3 \0 & 1 & | & 5\end{bmatrix}\]
The vertical line typically separates the coefficients from the constants, helping us visualize the boundary between the two parts. Working with augmented matrices allows us to apply row operations systematically, aiding in solving the system of equations efficiently.

Reduced Row-Echelon Form

Reduced row-echelon form (RREF) is a refined version of row-echelon form used to analyze and solve linear systems more neatly. A matrix is in reduced row-echelon form if:

• It first satisfies all the conditions of row-echelon form.
• Each leading 1 is the only non-zero entry in its column, meaning all other entries in the column containing a leading 1 should be zeros.

In the given example, the matrix \(\begin{bmatrix} 1 & 3 & -3 \ 0 & 1 & 5 \end{bmatrix}\) is not in reduced row-echelon form because the first row contains non-zero entries besides the leading 1 in the first column.
Achieving RREF often involves additional row operations such as:

• Replacing one row with a linear combination of itself and another row.
• Multiplying a row by a non-zero scalar.
• Adding multiples of one row to another row.

Such transformations ensure the matrix structure is uniform, making it easier to read off solutions directly when translating back to algebraic equations.

System of Equations

Systems of equations involve multiple equations that are solved simultaneously. Each equation in the system represents a linear relation involving the same set of variables.
For instance, consider again the system:\[\begin{aligned}& x + 3y = -3 \& y = 5\end{aligned}\]
A solution to the system of equations is a set of values for the variables that satisfy all equations simultaneously. For the above system, solving entails finding values for \(x\) and \(y\) such that both expressions hold true.
Systems can be solved in various ways:

• Substitution: solving one equation for a variable and then substituting that expression into another equation.
• Elimination: adding or subtracting equations to eliminate a variable, making it simpler to solve the rest.
• Matrix Methods: using augmented matrices and transforming them into a form where answers can be directly read off.

Understanding how to translate between matrices and systems of equations allows one to choose the most efficient method for any given problem, and with practice, you can solve even complex systems with confidence.