Row Operations and Augmented Matrices

Augmented Matrices and Row Operations

Learning Outcomes

• A matrix can represent and solve a system of equations.

• Express a system in matrix form by extracting coefficients and constants.

• Use a vertical line to separate coefficients from constants, creating an augmented matrix.

• Write the augmented matrix for a system of equations.

• Perform row operations on an augmented matrix.

Augmented Matrix

• A matrix form of a system of equations.

• Coefficients of variables and constants become entries of the matrix.

• A vertical line separates coefficient entries from constants (replaces equal signs).

Example: 2x2 System

• System of equations:
  $3x + 4y = 7$
  $4x - 2y = 5$

• Augmented matrix:
  $\begin{bmatrix} 3 & 4 & 7 \ 4 & -2 & 5 \end{bmatrix}$

Coefficient Matrix

• A matrix containing only the coefficients of the variables.

• Example:
  $\begin{bmatrix} 3 & 4 \ 4 & -2 \end{bmatrix}$

Example: 3x3 System

• System of equations:
  $3x - y - z = 0$
  $x + y = 5$
  $2x - 3z = 2$

• Coefficient matrix:

  $\begin{bmatrix} 3 & -1 & -1 \\ 1 & 1 & 0 \\ 2 & 0 & -3 \end{bmatrix}$
  

• Augmented matrix:
  $\begin{bmatrix} 3 & -1 & -1 & 0 \ 1 & 1 & 0 & 5 \ 2 & 0 & -3 & 2 \end{bmatrix}$ y, z terms).

• Equations must be in standard form: $ax + by + cz = d$

• Missing variable terms have a coefficient of 0.

How To: Write an Augmented Matrix

1. Write coefficients of x-terms down the first column.

2. Write coefficients of y-terms down the second column.

3. If there are z-terms, write coefficients down the third column.

4. Draw a vertical line and write the constants to the right.

Example: Writing the Augmented Matrix

• System of equations:
  $x + 2y - z = 3$
  $2x - y + 2z = 6$
  $x - 3y + 3z = 4$

• Augmented matrix:
  $\begin{bmatrix} 1 & 2 & -1 & 3 \ 2 & -1 & 2 & 6 \ 1 & -3 & 3 & 4 \end{bmatrix}$

Try It

• System of equations:
  $4x - 3y = 11$
  $3x + 2y = 4$

Writing a System of Equations from an Augmented Matrix

• Important to move between formats for smoother solutions.

• Use the information in an augmented matrix to write the system of equations in standard form.

Example: Writing a System of Equations

• Augmented matrix:
  $\begin{bmatrix} 1 & -3 & -5 & -2 \ 2 & -5 & -4 & 5 \ -3 & 5 & 4 & 6 \end{bmatrix}$

• System of equations:
  $x - 3y - 5z = -2$
  $2x - 5y - 4z = 5$
  $-3x + 5y + 4z = 6$

Try It

• Augmented matrix:
  $\begin{bmatrix} 1 & -1 & 1 & 5 \ 2 & -1 & 3 & 1 \ 0 & 1 & 1 & -9 \end{bmatrix}$

Row Operations

• Row operations: addition, multiplication by a constant, interchanging rows.

• Used to solve systems of equations.

• Convert the matrix to row-echelon form.

Row-Echelon Form

• Ones down the main diagonal (upper left to lower right).

• Zeros in every position below the main diagonal.

• Example:
  $\begin{bmatrix} 1 & a & b \ 0 & 1 & d \ 0 & 0 & 1 \end{bmatrix}$

Guidelines for Obtaining Row-Echelon Form

1. First nonzero number in any nonzero row is a 1 (leading 1).

2. All-zero rows are at the bottom.

3. Any leading 1 is below and to the right of a previous leading 1.

4. Any column containing a leading 1 has zeros in all other positions.

Row Operations Notation

1. Interchange rows: $Ri \leftrightarrow Rj$

2. Multiply a row by a constant: $cR_i$

3. Add the product of a row multiplied by a constant to another row: $Ri + cRj$

Gaussian Elimination

• Method to obtain row-echelon form.

• Goal: Write matrix A with 1s down the main diagonal and zeros below.

• $A = \begin{bmatrix} a{11} & a{12} & a{13} \ a{21} & a{22} & a{23} \ a{31} & a{32} & a{33} \end{bmatrix} \rightarrow \text{After Gaussian elimination} \rightarrow A = \begin{bmatrix} 1 & b{12} & b{13} \ 0 & 1 & b{23} \ 0 & 0 & 1 \end{bmatrix}$

• First step: Obtain a 1 as the first entry.

How To: Perform Row Operations to Achieve Row-Echelon Form

1. First equation should have a leading coefficient of 1 (interchange rows or multiply by a constant if needed).

2. Use row operations to obtain zeros down the first column below the first entry of 1.

3. Use row operations to obtain a 1 in row 2, column 2.

4. Use row operations to obtain zeros down column 2, below the entry of 1.

5. Use row operations to obtain a 1 in row 3, column 3.

6. Continue this process for all rows.

7. Place any all-zero rows at the bottom.

Example: Performing Row Operations on a 3x3 Augmented Matrix

• Given matrix:
  $\begin{bmatrix} 1 & -3 & 4 & 3 \ 3 & 2 & -5 & 6 \ -3 & 3 & 4 & 6 \end{bmatrix}$

• Step 1: The first row already has a 1 in row 1, column 1.

• Step 2: Multiply row 1 by -2 and add it to row 2, replace row 2 with the result:
  $-2R1 + R2 = R_2 \rightarrow \begin{bmatrix} 1 & -3 & 4 & 3 \ 0 & 8 & -13 & 0 \ -3 & 3 & 4 & 6 \end{bmatrix}$

• Step 3: Obtain a zero in row 3, column 1:
  $3R1 + R3 = R_3 \rightarrow \begin{bmatrix} 1 & -3 & 4 & 3 \ 0 & 11 & -17 & 15 \ 0 & -6 & 16 & 15 \end{bmatrix}$
  $\begin{bmatrix} 1 & -3 & 4 & 3 \ 0 & 1 & -2 & 0 \ 0 & -3 & 3 & 4 & 6 \end{bmatrix}$

• Step 4: Obtain a zero in row 3, column 2:
  $6R2 + R3 = R_3 \rightarrow \begin{bmatrix} 1 & -3 & 4 & 3 \ 0 & 1 & -2 & 0 \ 0 & 0 & 4 & 15 \end{bmatrix}$

• Step 5: Obtain a 1 in row 3, column 3:
  $\frac{1}{4}R3 = R3 \rightarrow \begin{bmatrix} 1 & -3 & 4 & 3 \ 0 & 1 & -2 & 0 \ 0 & 0 & 1 & \frac{15}{4} \end{bmatrix}$

Try It

• System of equations in row-echelon form:
  $x - 2y + 3z = 9$
  $-x + 3y = -4$
  $2x - 5y + 5z = 17$