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# IB Exam Review - Matrices

A Prezi Presentation by Brian Lo, Jeffrey Ma, Roy Cheung

#### Transcript of IB Exam Review - Matrices

Hey There! Welcome to our presentation on Matrices... This is an essential unit in IB Standard Level Math examinations We hope that you'll enjoy your stay on this walkthrough. Most importantly, have fun and relax! A little introduction about Matrices... What is a matrix essentially? Mathematical Definition: A rectangular Array of Numbers arranged in rows and columns. For example... we have here, a 3 by 3 matrix Note that a matrix consists of 'm' rows and 'n' columns

Therefore, m x n defines the order of a matrix How about the addition and subtraction of matrices? It's simple! The addition of 2 matrices must first require the 'same order' (As said previously) 'm x n'

You then add the corresponding 'elements' together! You're Done! And Subtraction.... again, the two matrices must be of same order, then proceed to subtract the corresponding elements! Alright! It's time to learn about Multiples of Matrices! Let 'q' represent a scalar, if a scalar is multiplied by a matrix, Matrix 'A' for example, the result is qA. Simply, you would multiply scalar 'q' by Matrix 'A'.

For example, given q = 1/2

and Matrix 'C' Side Note! Zero Matrices? Huh? Question, what is a zero matrix? It's when all elements within a matrix are zero! Caution! All zero matrices have this property... If 'A' is a matrix, given any order of course, and its corresponding matrix is a Zero Matrix, then A + 0 = 0 + A = A And now, onto Algebra! In the context of Matrices. A few general rules to consider... If 'A' and 'B' are matrices... A+B is also considered as a matrix.

Addition of A and B is equivalent to B and A

Matrix (A+B)+C = A+(B+C)

A+0 = 0+A = A (Did you remember this? We were just talking about this!)

A+(-A) = (-A)+A = 0

Do not put matrices as a fraction! Always consider it as a fraction multiplied to the matrix itself. Still don't understand? Or if you're bored, watch this! What do you say... we advance to Matrix Multiplication? All right! Off we goooooo!! More advanced matrices awaits you! Recall that a matrix is composed of rows and columns.

If you would like to multiply two matrices together, there must be certain guidelines to adhere to.

Given matrix 'A' with 'm' rows and 'n' columns

& Given matrix 'B' with 'o' rows and 'p' columns

The product of AB will contain 'm' rows and 'p' columns. Remember this! Check this out! Given matrix A

Consisting of 2 rows and 3 columns Given matrix B

Consisting of 3 rows and 1 column The result?

A 2 by 1 matrix!

Pay attention to how the elements of Matrix A are multiplied and added to its respective column and row!

Wait? How exactly? Here, let us clarify...

Multiply the elements found in the first row of the first matrix, (Matrix 'A' in this example!) by the first column of the second matrix (Matrix 'B' in this example.) You must then add all elements together to retrieve a final result. Proceed to do the same step with the second row of Matrix 'A'. You're complete! Now, for the IB examinations, you'll be expected to bring a calculator for paper 2. Let us teach you how to effectively solve matrices in a graphics calculator. dasd An example will suffice this portion of the presentation, here we go! The question requires you to multiply two matrixes A = B= 1st step: Go 2nd x-1, Scroll Right to insert the first Matrix as [A]

2nd step: Insert Matrix A, first insert the shape (3 x 2), and then insert the number in the Matrix 3rd step: Select the Matrix [A] by pressing 2nd X-1 and put a multiplication sign after it 4th step: Repeat step 1-3 with Matrix B 5th step: Select the Matrix [B] by pressing 2nd X-1 and pressing enter…the result appears! THERE YOU GO! Confusing you? Don't worry, we have all the matrix algebra guidelines to carry you through! Given that Matrix 'A' and Matrix 'B' are able to multiply, then the resulting 'AB' is also a matrix.

The addition of 'Matrix A' to 'Matrix B' can also be written as an addition of Matrix 'B' to Matrix 'A'.

Let O be a zero matrix. The multiplication of such matrix to a matrix 'A' for example, results in a zero matrix after all.

A(B+C) is equivalent to AB + AC

A 2x2 Matrix, Matrix 'A' for example, when multiplied with a corresponding 2x2 'Identity' Matrix. ex:

A matrix raised to the power of 'n', provided that 'n' is greater or equal to 2, can identify the matrix as a square and 'n' as an integer.

A worked example:

(2A + 4I)2

= (2A + 4I) (2A + 4I)

= (2A + 4I)2A + (A + 2I) 4I Rule: [B (C + D) = BC + BD]

= 4A2 + 8IA + 4AI + 8I2

= 4A2 + 8A + 4A + 8I Rule: [AI = IA = A] and [I2 = I]

= 4A2 + 12A + 8I

Let us now learn about the inverse of a 2 x 2 Matrix. However, you must first remember two rules when considering inverses.

1.) The multiplication of inverse of 'A', if it exists, satisfies

A-1A = AA-1 = I

2.) [A] has an inverse if |A| does not equal to 0

Rule 1 seems straight forward, but what does the second rule mean?

Well, |A| represents the determinant of a Matrix. And it is defined by

Imagine if ab – bc were to equal 0, then the formula becomes , which is undefined. This certainly adheres the the second guideline as mentioned. Now, to the main point, below is the formula for finding the inverse of a matrix: Given , a 2x2 matrix. The inverse of this will therefore be Here is a worked example to sort out the confusion... Find the Inverse of Matrix A, A = 1.)Change the matrix into the form , thereby turning it into

2.)Find the determinant of the Matrix using the form ad – bc

ad – bc = (3)(5) – (-6)(-2)

ad – bc = 15 – 12 = 3

3.) Therefore, the inverse of Matrix A is equal to = Pairs of Linear Equations - Let's Solve! Before we begin solving systems of equations, it is often a good idea to know if the system of equations has a unique solution. *a unique solution is where each variable has a single value*

So assume that our systems of equations is: To find out if we have a unique solution, we must know that the system of equations is solvable. By just looking at the systems of equations, we would not be able to predict if there is a unique solution so we must look at the matrix form.

To solve the systems of equations in matrix form, we must multiply both sides by the inverse of the 2x2 matrix. This means that if an inverse exists, there will be a unique solution. From what we have covered before, we know that

Det of 2x2 matrix = 4(8) -2(6) = 20

Therefore there is a unique solution for the systems of equations.

Now that we know we can solve the systems of equations we have to methods to solve it. The first method would be to recall algebraic substitution to solve the systems of equations.

The second method would be the one mentioned previously. Because we know that a matrix multiplied by its inverse causes the matrix to equate to one in matrix algebra, we can isolate the matrix with variables by multiplying both sides of the equation by the inverse of the 2x2 matrix.

From previous examples one can find the inverse from either technology or by multiplying the matrix by 1/determinant

The inverse of the 2x2 matrix is'

And by multiplying both sides of the equation by the matrix Which equates to x = 7/5 and y = 8/5 The Three by Three Determinant After we learned everything about 2x2 matrices, we can apply similar rules to the 3x3 matrices, but the major difference is calculating the determinant. The quickest way of explaining how to do so would be by using a general expression So for example if we had a matrix with numbers it would look something like this: So looking from that, it seems like finding the determinant of 3x3 matrices aren’t so bad! It’s just a matter of memorizing the formula. And if you are REALLY scared, never fear for anything harder than this will be dependant on your skills with the CALCULATOR! Finding the determinant can be just as easy

1) Enter your 3x3 matrix in the matrix edit section {(2nd) => (matrix) => (edit)}

2) Go to the math partition of the matrix section

3) Press the det button, then go back to the matrix section and select your matrix

4) Press enter and you will get your matrix

And now a small tutorial on how you should make use of a Graphing Calculator to find the inverse of a Three by Three Matrix! 1)Like anytime before put in your matrix

2)Select your matrix and bring it to the main screen

3)Then press the inverse button {x^-1}

4)Press {enter} and voila!

Lastly, we shall deal with unique solutions of a Three by Three Matrix... And here is where we apply everything we have learned.

In a previous section we used matrices algebra to solve a system of equations. We can also do this for a 3x3 matrix, with the help of our trusty friend the calculator.

The idea is still the same: if I let A rep my 3x3 matrix, B rep my 3x1 matrix, and C rep my 3x1 variable matrix I will have something that looks like this

A ∙C = B

Which can then be changed so that

A-1 A ∙C = A-1 B

Which then becomes

C = A-1 B

Using our trusty calculator, we plug in all the matrices and plug in the inverse of matrix 'A' and matrix 'B' into the calculator. Press enter and we have our answer!

YES! You have completed the review for Matrices. GOOD LUCK! Presentation by Brian Lo, Jeffrey Ma, Roy Cheung

Full transcriptTherefore, m x n defines the order of a matrix How about the addition and subtraction of matrices? It's simple! The addition of 2 matrices must first require the 'same order' (As said previously) 'm x n'

You then add the corresponding 'elements' together! You're Done! And Subtraction.... again, the two matrices must be of same order, then proceed to subtract the corresponding elements! Alright! It's time to learn about Multiples of Matrices! Let 'q' represent a scalar, if a scalar is multiplied by a matrix, Matrix 'A' for example, the result is qA. Simply, you would multiply scalar 'q' by Matrix 'A'.

For example, given q = 1/2

and Matrix 'C' Side Note! Zero Matrices? Huh? Question, what is a zero matrix? It's when all elements within a matrix are zero! Caution! All zero matrices have this property... If 'A' is a matrix, given any order of course, and its corresponding matrix is a Zero Matrix, then A + 0 = 0 + A = A And now, onto Algebra! In the context of Matrices. A few general rules to consider... If 'A' and 'B' are matrices... A+B is also considered as a matrix.

Addition of A and B is equivalent to B and A

Matrix (A+B)+C = A+(B+C)

A+0 = 0+A = A (Did you remember this? We were just talking about this!)

A+(-A) = (-A)+A = 0

Do not put matrices as a fraction! Always consider it as a fraction multiplied to the matrix itself. Still don't understand? Or if you're bored, watch this! What do you say... we advance to Matrix Multiplication? All right! Off we goooooo!! More advanced matrices awaits you! Recall that a matrix is composed of rows and columns.

If you would like to multiply two matrices together, there must be certain guidelines to adhere to.

Given matrix 'A' with 'm' rows and 'n' columns

& Given matrix 'B' with 'o' rows and 'p' columns

The product of AB will contain 'm' rows and 'p' columns. Remember this! Check this out! Given matrix A

Consisting of 2 rows and 3 columns Given matrix B

Consisting of 3 rows and 1 column The result?

A 2 by 1 matrix!

Pay attention to how the elements of Matrix A are multiplied and added to its respective column and row!

Wait? How exactly? Here, let us clarify...

Multiply the elements found in the first row of the first matrix, (Matrix 'A' in this example!) by the first column of the second matrix (Matrix 'B' in this example.) You must then add all elements together to retrieve a final result. Proceed to do the same step with the second row of Matrix 'A'. You're complete! Now, for the IB examinations, you'll be expected to bring a calculator for paper 2. Let us teach you how to effectively solve matrices in a graphics calculator. dasd An example will suffice this portion of the presentation, here we go! The question requires you to multiply two matrixes A = B= 1st step: Go 2nd x-1, Scroll Right to insert the first Matrix as [A]

2nd step: Insert Matrix A, first insert the shape (3 x 2), and then insert the number in the Matrix 3rd step: Select the Matrix [A] by pressing 2nd X-1 and put a multiplication sign after it 4th step: Repeat step 1-3 with Matrix B 5th step: Select the Matrix [B] by pressing 2nd X-1 and pressing enter…the result appears! THERE YOU GO! Confusing you? Don't worry, we have all the matrix algebra guidelines to carry you through! Given that Matrix 'A' and Matrix 'B' are able to multiply, then the resulting 'AB' is also a matrix.

The addition of 'Matrix A' to 'Matrix B' can also be written as an addition of Matrix 'B' to Matrix 'A'.

Let O be a zero matrix. The multiplication of such matrix to a matrix 'A' for example, results in a zero matrix after all.

A(B+C) is equivalent to AB + AC

A 2x2 Matrix, Matrix 'A' for example, when multiplied with a corresponding 2x2 'Identity' Matrix. ex:

A matrix raised to the power of 'n', provided that 'n' is greater or equal to 2, can identify the matrix as a square and 'n' as an integer.

A worked example:

(2A + 4I)2

= (2A + 4I) (2A + 4I)

= (2A + 4I)2A + (A + 2I) 4I Rule: [B (C + D) = BC + BD]

= 4A2 + 8IA + 4AI + 8I2

= 4A2 + 8A + 4A + 8I Rule: [AI = IA = A] and [I2 = I]

= 4A2 + 12A + 8I

Let us now learn about the inverse of a 2 x 2 Matrix. However, you must first remember two rules when considering inverses.

1.) The multiplication of inverse of 'A', if it exists, satisfies

A-1A = AA-1 = I

2.) [A] has an inverse if |A| does not equal to 0

Rule 1 seems straight forward, but what does the second rule mean?

Well, |A| represents the determinant of a Matrix. And it is defined by

Imagine if ab – bc were to equal 0, then the formula becomes , which is undefined. This certainly adheres the the second guideline as mentioned. Now, to the main point, below is the formula for finding the inverse of a matrix: Given , a 2x2 matrix. The inverse of this will therefore be Here is a worked example to sort out the confusion... Find the Inverse of Matrix A, A = 1.)Change the matrix into the form , thereby turning it into

2.)Find the determinant of the Matrix using the form ad – bc

ad – bc = (3)(5) – (-6)(-2)

ad – bc = 15 – 12 = 3

3.) Therefore, the inverse of Matrix A is equal to = Pairs of Linear Equations - Let's Solve! Before we begin solving systems of equations, it is often a good idea to know if the system of equations has a unique solution. *a unique solution is where each variable has a single value*

So assume that our systems of equations is: To find out if we have a unique solution, we must know that the system of equations is solvable. By just looking at the systems of equations, we would not be able to predict if there is a unique solution so we must look at the matrix form.

To solve the systems of equations in matrix form, we must multiply both sides by the inverse of the 2x2 matrix. This means that if an inverse exists, there will be a unique solution. From what we have covered before, we know that

Det of 2x2 matrix = 4(8) -2(6) = 20

Therefore there is a unique solution for the systems of equations.

Now that we know we can solve the systems of equations we have to methods to solve it. The first method would be to recall algebraic substitution to solve the systems of equations.

The second method would be the one mentioned previously. Because we know that a matrix multiplied by its inverse causes the matrix to equate to one in matrix algebra, we can isolate the matrix with variables by multiplying both sides of the equation by the inverse of the 2x2 matrix.

From previous examples one can find the inverse from either technology or by multiplying the matrix by 1/determinant

The inverse of the 2x2 matrix is'

And by multiplying both sides of the equation by the matrix Which equates to x = 7/5 and y = 8/5 The Three by Three Determinant After we learned everything about 2x2 matrices, we can apply similar rules to the 3x3 matrices, but the major difference is calculating the determinant. The quickest way of explaining how to do so would be by using a general expression So for example if we had a matrix with numbers it would look something like this: So looking from that, it seems like finding the determinant of 3x3 matrices aren’t so bad! It’s just a matter of memorizing the formula. And if you are REALLY scared, never fear for anything harder than this will be dependant on your skills with the CALCULATOR! Finding the determinant can be just as easy

1) Enter your 3x3 matrix in the matrix edit section {(2nd) => (matrix) => (edit)}

2) Go to the math partition of the matrix section

3) Press the det button, then go back to the matrix section and select your matrix

4) Press enter and you will get your matrix

And now a small tutorial on how you should make use of a Graphing Calculator to find the inverse of a Three by Three Matrix! 1)Like anytime before put in your matrix

2)Select your matrix and bring it to the main screen

3)Then press the inverse button {x^-1}

4)Press {enter} and voila!

Lastly, we shall deal with unique solutions of a Three by Three Matrix... And here is where we apply everything we have learned.

In a previous section we used matrices algebra to solve a system of equations. We can also do this for a 3x3 matrix, with the help of our trusty friend the calculator.

The idea is still the same: if I let A rep my 3x3 matrix, B rep my 3x1 matrix, and C rep my 3x1 variable matrix I will have something that looks like this

A ∙C = B

Which can then be changed so that

A-1 A ∙C = A-1 B

Which then becomes

C = A-1 B

Using our trusty calculator, we plug in all the matrices and plug in the inverse of matrix 'A' and matrix 'B' into the calculator. Press enter and we have our answer!

YES! You have completed the review for Matrices. GOOD LUCK! Presentation by Brian Lo, Jeffrey Ma, Roy Cheung