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Matrices

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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.

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

- 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.

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.

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

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

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 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 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.

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

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

- This forms a 2 x 1 matrix

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

- This forms a 2 x 1 matrix

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

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

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

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

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

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

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

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

- Solve the following equation:

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

- Solve the following equation:

- Solve the following equation: cake and Bill bought 2 pies, a drink and a cake.

- Solve the following equation: cake and Bill bought 2 pies, a drink and a cake.

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 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.”

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

- Solve the following

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

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

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.”

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

- Solve the following

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

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

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