Chapter 6 Notes: Linear Equations and Matrix Algebra

Chapter 6: Linear Equations and Matrix Algebra

Sections 6.1 & 6.2: The All Integer Method

Method 1: Solving Graphically

• Consistent, Independent: One solution.

• Inconsistent, Independent: No solution.

• Consistent, Dependent: Infinite number of solutions (y depends on x).

Method 2: Solving by Substitution

Method 3: Solving by Elimination

Method 4: Solving by The All Integer Echelon Method

The All Integer Method

• First Step: Create the Setup (initial) table.

• Last Step: The table should result to the identity matrix I where $x$, $y$, and $z$ equals a constant.

• Answer Format: $x = a$, $y = b$, $z = c$

All Integer Method Steps

1. Setup Initial Table: Select the first Pivot Element (the first element in the table).

2. Copy Pivot Row: Make all other elements in the pivot column = 0.

3. Replace Other Elements: Use the "criss-cross" multiplication method.

Criss-Cross Operation

• Current Pivot: The element being used as the pivot.

• Previous Pivot: The pivot from the previous tableau (if none, assume = 1).

• The "criss-cross" method involves creating a rectangle with the pivot element and the element to be replaced at facing corners.

• $New Element = \frac{(Pivot \times ElementToBeReplaced) - (ProductOfOppositeDiagonal)}{(PreviousPivot)}$

• Note: The result must be an Integer until the very last step.

First Tableau, First Pivot

• After performing the criss-cross operation, update the table with the new values.

Second Tableau, Second Pivot

• Select a new pivot element diagonally in the next row.

• New (Second) Pivot = 4

• Previous Pivot = 2

• Copy the pivot row and make all other elements in the pivot column = 0.

• Note: For columns already pivoted, the old pivot will change to the new and current pivot and the 0's will stay.

• Replace the other elements using the "criss-cross" multiplication method.

Third Tableau, Third Pivot

• Select a new pivot element diagonally in the next row.

• New (Third) Pivot = -4

• Previous Pivot = 4

• Copy the pivot row and make all other elements in the pivot column = 0.

• Note: For columns already pivoted, the old pivot will change to the new and current pivot and the 0's will stay.

• Replace the other elements using the "criss-cross" multiplication method.

Last Tableau

• Select a new pivot element diagonally in the next row.

• New Pivot = No more rows.

• No more pivot.

• Previous or last Pivot = -4.

• Divide all elements by the last pivot which is = -4.

• This is the last step and the only step where you can get fractions or decimals as answers.

Special Cases

Case 1: No Solution

• If any time there is a row with all 0's to the left and a non-zero to the right, the system is inconsistent (no solution).

Case 2: Infinite Number of Solutions

• Eliminate the line of all 0's.

• Move to the next pivot if possible.

• Translate the solution. It is Linearly Dependent.

• $x=2-3z$

• $y=3-5z$

• z = any #

Section 6.3: Matrix Notation

Matrices

• A matrix is a rectangular or square array of values arranged in rows and columns.

• An $m \times n$ matrix A has m rows and n columns.

Equality of Matrices

• Two matrices are equal if they have the same dimension and the same elements.

Addition and Subtraction of Matrices

• Matrices A and B must have the same dimensions.

Scalar Product

• If $k$ is a real number, then the scalar product $k \cdot A$ is obtained by multiplying each element of $A$ by $k$.

Transpose of a Matrix

• Each row becomes a column, and each column becomes a row.

Matrix Multiplication

• Multiply each Row in matrix A by each Column in matrix B.

• Multiply corresponding entries and then add the resulting products.

Matrix Multiplication Dimensions

• If A is a $m \times n$ matrix and B is a $n \times p$ matrix, then A.B is a $m \times p$ matrix. The inner dimensions (n) must match for multiplication to be possible.

Section 6.3 Cont.: Inverse Matrix

• To find the inverse matrix of A using the All Integer Method:

  • Step 1: Re-write it with the Identity Matrix I next to it on the right side.

• (The Identity Matrix I: the square matrix where all Diagonal elements = 1, the rest are zeros)

  • Step 2: Do the pivot steps (2 pivots for two rows), and the last step should be:

  • Step 3: The Identity Matrix I is now on the left side, and the Inverse Matrix $A^{-1}$ is on the right side.

• You can check your answer by multiplying the original matrix A and the inverse $A^{-1}$. The answer must be an Identity Matrix I.

• Note: Not every matrix has an inverse.