# Matrices - PowerPoint PPT Presentation

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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

### 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

• 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:

• 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:

• 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

### Equality:

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

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

### 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.

### Scalar Multiplication

• Multiplying by a number.

• Each element is multiplied by the number.

• Also

### Zero Matrix

• Every element in the matrix is zero.

• This is the identity for addition.

• Example:

### Identity or unit matrix

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

• Example:

### 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 Multiplication

• This forms a 2 x 1 matrix

### Matrix Multiplication

• This forms a 2 x 1 matrix

### Remember

• A linear equation with 2 variables is a line

• E.g.

### 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 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

• 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 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

• 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

• 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

### 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:

• Solve the following equation:

### Write in matrix form:

• Solve the following equation:

• Solve the following equation:

• Solve the following equation:

### Solution is (1, 2)

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

### Scenario 2

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

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

### Example:

• Solve the following

### 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.”

### Example:

• Solve the following