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


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

  • 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:

  • 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


Order of a matrix


Equality:

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


Addition

  • 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.


Example


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.


A linear equation with 3 variables is a plane.


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


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


Step 3: Divide (2) by -3


The matrix is now in row echelon form


Using back substitution to solve:


We could have done 1 more step


This is called reduced row echelon form


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


Write down the augmented matrix


The system is “independent and inconsistent and has nosolution.”


The lines are parallel and hence there is no solution


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


Write down the augmented matrix


The system is “dependentand consistent and has infinitesolutions.”


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


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