Matrices
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Matrices. Revision: Substitution. Solve for one variable in one of the equations. Substitute this expression into the other equation to get one equation with one unknown. Back substitute the value found in step 2 into the expression from step 1. Check. Solve:.

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Revision substitution
Revision: Substitution

  • Solve for one variable in one of the equations.

  • Substitute this expression into the other equation to get one equation with one unknown.

  • Back substitute the value found in step 2 into the expression from step 1.

  • Check.


Solve:

Step 1: Solve equation 1 for y.

Step 2: Substitute this expression into eqn 2 for y.

Step 3: Solve for x.

Step 4: Substitute this value for x into eqn 1 and find the corresponding y value.

Step 5: Check in both equations.


Elimination
Elimination

  • Adjust the coefficients.

  • Add the equations to eliminate one of the variable. Then solve for the remaining variable.

  • Back substitute the value found in step 2 into one of the original equations to solve for the other variable.

  • Check.


Solve:

Step1: The coefficients on the y variables are opposites. No multiplication is needed.

Step 2: Add the two equations together.

Step 3: Solve for x.

Step 4: Substitute this value for x into either eqn and find the corresponding y value.

Step 5: Check in both equations.


Definition
Definition:

  • In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

  • In simple terms, a table of numbers.


Example
Example:

  • John and Bill went to the tuck shop. John bought a pie and cake and Bill bought 2 pies, a drink and a cake.


John and Bill went to the tuck shop. John bought a pie and cake and Bill bought 2 pies, a drink and a cake.

Matrix


Basics for matrices
Basics for matrices cake and Bill bought 2 pies, a drink and a cake.


Order of a matrix
Order of a matrix cake and Bill bought 2 pies, a drink and a cake.


Equality
Equality: cake and Bill bought 2 pies, a drink and a cake.

  • Two matrices are equal only if they have the same order and their corresponding elements are equal.


Addition
Addition cake and Bill bought 2 pies, a drink and a cake.

  • Two matrices must be of the same order before they can be added.


Subtraction
Subtraction cake and Bill bought 2 pies, a drink and a cake.

  • The negative matrix is obtained by taking the opposite value of each element in the matrix.

Subtraction is the same as adding the negative matrix.


Example1
Example cake and Bill bought 2 pies, a drink and a cake.


Scalar multiplication
Scalar Multiplication cake and Bill bought 2 pies, a drink and a cake.

  • Multiplying by a number.

  • Each element is multiplied by the number.

  • Also


Zero matrix
Zero Matrix cake and Bill bought 2 pies, a drink and a cake.

  • Every element in the matrix is zero.

  • This is the identity for addition.

  • Example:


Identity or unit matrix
Identity or unit matrix cake and Bill bought 2 pies, a drink and a cake.

  • The elements of the leading diagonal are all 1 and all other elements are zero.

  • Example:


Matrix multiplication
Matrix Multiplication cake and Bill bought 2 pies, a drink and a cake.

  • Two matrices can be multiplied if the number of columns of the first matrix equals the number of rows of the second matrix.

  • Example:


Matrix multiplication1
Matrix Multiplication cake and Bill bought 2 pies, a drink and a cake.

  • This forms a 2 x 1 matrix


Matrix multiplication2
Matrix Multiplication cake and Bill bought 2 pies, a drink and a cake.

  • This forms a 2 x 1 matrix


Remember
Remember cake and Bill bought 2 pies, a drink and a cake.

  • A linear equation with 2 variables is a line

  • E.g.


A linear equation with 3 variables is a plane
A linear equation with 3 variables is a plane cake and Bill bought 2 pies, a drink and a cake..


Solving problems using matrices
Solving problems using matrices cake and Bill bought 2 pies, a drink and a cake.

  • Scenario 1

  • Two lines intersect at one point. There is a unique solution.

  • This means the system is independent and consistent with a uniquesolution.


Solving problems using matrices1
Solving problems using matrices cake and Bill bought 2 pies, a drink and a cake.

  • Two small jugs and one large jug can hold 8 cups of water.  One large jug minus one small jug constitutes 2 cups of water.  How many cups of  water can each jug hold?


Step 1 write down the equations
Step 1: write down the equations cake and Bill bought 2 pies, a drink and a cake.

  • Two small jugs and one large jug can hold 8 cups of water.  One large jug minus one small jug constitutes 2 cups of water.  How many cups of  water can each jug hold?

  • Let x = number of cups of water that the large jug holds

  • Let y = number of cups of water that the small jug holds


Step 1 write down the equations1
Step 1: write down the equations cake and Bill bought 2 pies, a drink and a cake.

  • Two small jugs and one large jug can hold 8 cups of water.  One large jug minus one small jug constitutes 2 cups of water.  How many cups of  water can each jug hold?

  • Let x = number of cups of water that the small jug holds

  • Let y = number of cups of water that the large jug holds


Step 2 create the matrix
Step 2: Create the matrix cake and Bill bought 2 pies, a drink and a cake.

  • Two small jugs and one large jug can hold 8 cups of water.  One large jug minus one small jug constitutes 2 cups of water.  How many cups of  water can each jug hold?

  • Let x = number of cups of water that the small jug holds

  • Let y = number of cups of water that the large jug holds


This is called the augmented matrix of the system
This is called the augmented matrix of the system cake and Bill bought 2 pies, a drink and a cake.

  • Two small jugs and one large jug can hold 8 cups of water.  One large jug minus one small jug constitutes 2 cups of water.  How many cups of  water can each jug hold?

  • Let x = number of cups of water that the small jug holds

  • Let y = number of cups of water that the large jug holds


Step 2 k eep equation 1 and eliminate x from 2
Step 2: cake and Bill bought 2 pies, a drink and a cake.Keep equation (1) and eliminate ‘x’ from (2)


Step 3 divide 2 by 3
Step 3: Divide (2) by -3 cake and Bill bought 2 pies, a drink and a cake.


The matrix is now in row echelon form
The matrix is now in cake and Bill bought 2 pies, a drink and a cake.row echelon form


Using back substitution to solve
Using back substitution to solve: cake and Bill bought 2 pies, a drink and a cake.


We could have done 1 more step
We could have done 1 more step cake and Bill bought 2 pies, a drink and a cake.


This is called reduced row echelon form
This is called cake and Bill bought 2 pies, a drink and a cake.reduced row echelon form


We have a unique solution 2 2
We have a unique solution (2, 2) cake and Bill bought 2 pies, a drink and a cake.

  • This means the system is independent and consistent with a uniquesolution.

  • Geometrically: The lines intersect at a unique point because the gradients are not the same.


Example 2
Example 2: cake and Bill bought 2 pies, a drink and a cake.

  • Solve the following equation:


Write in matrix form
Write in matrix form: cake and Bill bought 2 pies, a drink and a cake.

  • Solve the following equation:




Solution is 1 2
Solution is (1, 2) cake and Bill bought 2 pies, a drink and a cake.

  • This means the system is independent and consistent with a uniquesolution.


Scenario 2
Scenario 2 cake and Bill bought 2 pies, a drink and a cake.

  • The two lines don’t intersect because the lines are parallel.

  • The system is “independentand inconsistent and has nosolution.”


Example2
Example: cake and Bill bought 2 pies, a drink and a cake.

  • Solve the following


Write down the augmented matrix
Write down the augmented matrix cake and Bill bought 2 pies, a drink and a cake.


The system is independent and inconsistent and has no solution
The system is cake and Bill bought 2 pies, a drink and a cake.“independent and inconsistent and has nosolution.”


The lines are parallel and hence there is no solution
The lines are parallel and hence there is no solution cake and Bill bought 2 pies, a drink and a cake.


Scenario 3
Scenario cake and Bill bought 2 pies, a drink and a cake.3

Identical lines which intersect at an infinite number of points.

The system is “dependent and consistent and has an infinite number of solutions.”


Example3
Example: cake and Bill bought 2 pies, a drink and a cake.

  • Solve the following


Write down the augmented matrix1
Write down the augmented matrix cake and Bill bought 2 pies, a drink and a cake.


The system is dependent and consistent and has infinite solutions
The system is cake and Bill bought 2 pies, a drink and a cake.“dependentand consistent and has infinitesolutions.”



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