Lesson 2 Introduction to Matrices

2.1 Lecture Content

Video: Using Matrices to Solve Systems

Video: Applications of Linear Systems

2.2 Lecture Notes

2.2.1 Augmented Matrix

An augmented matrix is a compact representation of a system of linear equations, showing only the coefficients and constants. To create one:

  1. Write the coefficients of the variables from each equation into rows.
  2. Add a vertical line (conceptually) separating the coefficients from the constants on the right-hand side.

Example:

For the system \[\begin{align*}x + 2y &= 5\\ 3x - y &= 4 \end{align*}\]

The augmented matrix is:

\[ \begin{bmatrix} 1 & 2 & | & 5 \\ 3 & -1 & | & 4 \end{bmatrix} \]

2.2.2 Elementary Row Operations

There are three types of elementary row operations used in row reduction:

  1. Row Swapping (Ri ↔︎ Rj)
    Swap two rows.

  2. Row Scaling (k·Ri)
    Multiply a row by a non-zero constant.

  3. Row Replacement (Ri + k·Rj → Ri)
    Add or subtract a multiple of one row to another row.

These operations are used to transform a matrix into a simpler form without changing the solution to the system.

2.2.3 Gauss-Jordan Elimination Steps

Gauss-Jordan Elimination is a method to solve a system of linear equations by transforming the augmented matrix into reduced row echelon form (RREF).

Steps:

  1. Write the augmented matrix for the system.
  2. Use elementary row operations to get a leading 1 (pivot) in the top-left corner.
  3. Create zeros below and above the pivot.
  4. Move to the next row and repeat the process for the next pivot (diagonal).
  5. Continue until you have 1s down the diagonal and 0s elsewhere in the coefficient part.
  6. Read the solution directly from the matrix.

Goal:
Transform the matrix into:

\[ \begin{bmatrix} 1 & 0 & \cdots & 0 & | & a \\ 0 & 1 & \cdots & 0 & | & b \\ \vdots & \vdots & \ddots & \vdots & | & \vdots \\ 0 & 0 & \cdots & 1 & | & c \end{bmatrix} \]

Which corresponds to the solution \(x = a\), \(y = b\), etc.

2.3 Examples

Video: Solutions

  1. When is a matrix in Row Echelon Form?

  2. Is this matrix in row echelon form? \[ \begin{bmatrix} 5 & 6 \\ 0 & 2\\ 0&0 \end{bmatrix} \]

  3. Is this matrix in row echelon form? \[ \begin{bmatrix} 5 & -5 & -4 &|& 1 \\ 3 & 0 & -10 &|& -2\\ 0 & 0 & 2 &|& -3 \end{bmatrix} \]

  4. Find the augmented coefficient matrix corresponding to the system:

\[ \begin{align*} -3x + 4 &= 4y \\ -9x - 6y &= -4 \\ 2x + 1 &= 5y \end{align*} \]

  1. Find the reduced row echelon form (RREF) of this augmented matrix: \[ \begin{bmatrix} -2 & 0 & 5&|&250\\ 10 & 1 & -26&|&400\\ 3 & 0 & -8&|&150 \end{bmatrix} \]

  2. Solve the system by row reduction: \[ \begin{align*} x_1 + 3x_2 + 3x_3 + &= -6 \\ -x_1 - 2x_2 - 6x_3 &= -5 \\ x_1 + 3x_2 + 4x_3 &= 5 \end{align*} \]

  3. Solve the system by row reduction: \[ \begin{align*} 7w - 16x + 3y + 8z &=-6\\ w + 3x + 2y &= 4\\ 2w - 4x + y + 2 &= 1\\ -2w + 3x - y - z &= 4 \end{align*} \]

  4. Given the following augmented matrix, give a parameterized set of solutions for the system. \[ \begin{bmatrix} 14 & -8 & 2 &|& 128\\ 2 & -1 & 1 &|& 17 \end{bmatrix} \]

2.4 Practice Problems

  1. Use Gauss-Jordan to solve:
    \[ \begin{cases} 2x + y = 2 \\ -2x + y = 2 \end{cases} \]

  2. Use Gauss-Jordan to solve:
    \[ \begin{cases} 2x - 3y = 1 \\ 6x - 9y = 3 \end{cases} \]

  3. Use Gauss-Jordan to solve:
    \[ \begin{cases} x + y = 1 \\ 3x - 2y = -1 \\ 5x - y = \frac{1}{5} \end{cases} \]

  4. Use Gauss-Jordan to solve:
    \[ \begin{cases} x - y + z - u + v = 1 \\ y + z + u + v = 2 \\ z - u + v = 1 \\ u + v = 1 \\ v = 1 \end{cases} \]

  5. Use Gauss-Jordan to solve:
    \[ \begin{cases} x + 2y + 3z + 4w + t = 6 \\ 2x + 3y + 4z + 5w + t = 5 \\ 3x + 4y + 5z + w + 2t = 4 \\ 4x + 5y + z + 2w + 3t = 3 \\ 5x + y + 2z + 3w + 4t = 2 \end{cases} \]

  6. (APPLIED) You own a hamburger franchise and want to use leftover ingredients to make burgers.

    • Plain burgers: 1 beef patty, 1 bread roll
    • Double cheeseburgers: 2 beef patties, 1 bread roll, 2 cheese slices
    • Regular cheeseburgers: 1 beef patty, 1 bread roll, 1 cheese slice
      You have 13 bread rolls, 19 beef patties, and 15 cheese slices. How many of each type can you make?

2.5 Self-Assessment

Time yourself and solve within 20 minutes:

  1. Use Gauss-Jordan to solve:
    \[ \begin{cases} 4x - 2y = 1 \\ -2x + y = 4 \end{cases} \]
  2. Use Gauss-Jordan to solve:
    \[ \begin{cases} \frac{-2x}{3} + \frac{y}{2} = \frac{-1}{6} \\ \frac{x}{4} - y = \frac{-3}{4} \end{cases} \]
  3. Explain why \(3R_3 - 2R_1 \to R_3\) is not an elementary row operation.
  4. In 2007, total revenues from sales of country music, children’s music, and soundtracks were $1.5 billion. Country music brought in 12 times soundtracks, children’s music 3 times soundtracks. How much revenue for each?

2.6 Lesson Checklist

This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.

\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]