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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Matrices

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Matrices

Matrices


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.


Matrices

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.


Matrices

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.


Matrices

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


Order of a matrix

Order of a matrix


Equality

Equality:

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


Addition

Addition

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


Subtraction

Subtraction

  • 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


Scalar multiplication

Scalar Multiplication

  • Multiplying by a number.

  • Each element is multiplied by the number.

  • Also


Zero matrix

Zero Matrix

  • Every element in the matrix is zero.

  • This is the identity for addition.

  • Example:


Identity or unit matrix

Identity or unit matrix

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

  • Example:


Matrix multiplication

Matrix Multiplication

  • 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

  • This forms a 2 x 1 matrix


Matrix multiplication2

Matrix Multiplication

  • This forms a 2 x 1 matrix


Remember

Remember

  • 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.


Solving problems using matrices

Solving problems using matrices

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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: Keep equation (1) and eliminate ‘x’ from (2)


Step 3 divide 2 by 3

Step 3: Divide (2) by -3


The matrix is now in row echelon form

The matrix is now in row echelon form


Using back substitution to solve

Using back substitution to solve:


We could have done 1 more step

We could have done 1 more step


This is called reduced row echelon form

This is called reduced row echelon form


We have a unique solution 2 2

We have a unique solution (2, 2)

  • 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:

  • Solve the following equation:


Write in matrix form

Write in matrix form:

  • Solve the following equation:


Matrices

  • Solve the following equation:


Matrices

  • Solve the following equation:


Solution is 1 2

Solution is (1, 2)

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


Scenario 2

Scenario 2

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

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


Example2

Example:

  • Solve the following


Write down the augmented matrix

Write down the augmented matrix


The system is independent and inconsistent and has no solution

The system is “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


Scenario 3

Scenario 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:

  • Solve the following


Write down the augmented matrix1

Write down the augmented matrix


The system is dependent and consistent and has infinite solutions

The system is “dependentand consistent and has infinitesolutions.”


The lines are identical and hence there is an infinite number of solutions

The lines are identical and hence there is an infinite number of solutions


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