Problem 9 - 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. $$\left[\begin{array}{llll}{1} & {2} & {8} & {0} \\ {0} & {1} & {3} & {2} \\\ {0} & {0} & {0} & {0}\end{array}\right]$$

Step-by-Step Solution

Answer

(a) Yes, it's in row-echelon form. (b) Yes, it's in reduced row-echelon form. (c) The system is \(x + 2y + 8z = 0\) and \(y + 3z = 2\).

Step 1: Identify Non-zero Rows for Row-Echelon Form

To check if a matrix is in row-echelon form, we first identify non-zero rows. In this matrix:\[\begin{bmatrix}1 & 2 & 8 & 0 \0 & 1 & 3 & 2 \0 & 0 & 0 & 0\end{bmatrix}\]The first two rows are non-zero, and the third row is all zeros.

Step 2: Check Leading Entries Positions

In row-echelon form, each leading entry (the first non-zero number from the left in a row) of a non-zero row must be to the right of the leading entry of the row directly above it. In this matrix: - The leading entry of the first row is 1, in the first column. - The leading entry of the second row is 1, in the second column. - These are consistent with row-echelon form.

Step 3: Verify Downward Zero Rows

In row-echelon form, any zero rows must be at the bottom of the matrix. The bottom row of this matrix is all zeros, which is consistent with row-echelon form.

Step 4: Conclusion on Row-Echelon Form

Based on Steps 1 to 3, the matrix satisfies the criteria for row-echelon form. Therefore, the matrix is in row-echelon form.

Step 5: Check for Leading Entry as 1 for Reduced Row-Echelon Form

In reduced row-echelon form, the leading entry in each non-zero row must be 1, which is already true for this matrix as the leading entries in each row are 1.

Step 6: Ensure Column Cleanliness for Reduced Row-Echelon Form

For reduced row-echelon form, each leading 1 must be the only non-zero entry in its column. In this matrix, each leading column (containing a leading 1) does not have any other non-zero entries: - The first column has a leading 1 only in the first row. - The second column has a leading 1 only in the second row. - Since these columns meet the requirement, this matrix is also in reduced row-echelon form.

Step 7: Write the System of Equations from Augmented Matrix

Extract the system of equations from the augmented matrix by interpreting each row as an equation:- First row: \(1x + 2y + 8z = 0\)- Second row: \(0x + 1y + 3z = 2\)- Third row is a zero row, providing no information.Thus, the system of equations is:\[\begin{aligned}&x + 2y + 8z = 0 \&y + 3z = 2\end{aligned}\]

Key Concepts

Reduced Row Echelon Form

The Reduced Row Echelon Form (RREF) of a matrix goes a step further than row-echelon form by requiring additional conditions for simplification. In RREF, each leading entry (also called a pivot) must be a 1, which simplifies handling systems of equations. This form is characterized by the following:

• All nonzero rows in the matrix are above any rows of all zeros.
• Each leading 1 is to the right of leading ones in rows above.
• Each leading 1 is the only nonzero entry in its column.

For the matrix \(\begin{bmatrix} 1 & 2 & 8 & 0 \ 0 & 1 & 3 & 2 \ 0 & 0 & 0 & 0 \end{bmatrix}\), it satisfies the conditions for RREF:

• Both non-zero rows have leading entries of 1.
• The columns containing leading ones do not have any other nonzero numbers.

This full reduction makes solving systems of equations straightforward, as it clearly highlights the solutions and dependencies.

System of Equations

Systems of equations are collections of multiple equations that share variables. Their solutions are the set of variable values that satisfy all the equations simultaneously. Often, matrices are used to represent systems because they simplify calculations and manipulations.

• When viewing a matrix as a system of equations, each row corresponds to an equation.
• The coefficients of variables appear as matrix entries.
• Any solutions must satisfy each equation in the system.

Given the matrix \[\begin{bmatrix} 1 & 2 & 8 & 0 \ 0 & 1 & 3 & 2 \ 0 & 0 & 0 & 0 \end{bmatrix}\]this can be interpreted as the system of equations:

• \(x + 2y + 8z = 0\)
• \(y + 3z = 2\)

The third row being all zeros implies it adds no constraints, which simplifies the system, focusing on fewer active equations.

Augmented Matrix

An augmented matrix is a concise way to represent a system of linear equations, including the constants on the right side of the equations. It combines the coefficient matrix with a column for the constants from the equations' right-hand sides.

• This setup allows for efficient computation and row operations.
• The row operations aim to transform the matrix into a simpler form, such as row-echelon or reduced row-echelon form.

For the system:

• \(x + 2y + 8z = 0\)
• \(y + 3z = 2\)

The augmented matrix form is expressed as:\[\begin{bmatrix} 1 & 2 & 8 & | & 0 \ 0 & 1 & 3 & | & 2 \ 0 & 0 & 0 & | & 0 \end{bmatrix}\]where the vertical line separates the coefficient part of the matrix from the constants, clarifying which part of each equation corresponds to which role in the matrix.