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How to Multiply Matrices and Solve Equations: Step by Step for Kids

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How to Multiply Matrices and Solve Equations: Step by Step for Kids
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Ahmed Nour ✓™

@ahmednour

·

412 Followers

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This document covers matrix operations, determinants, and inverse matrices, with a focus on solving systems of equations using matrix methods. It provides step-by-step explanations and examples for various matrix calculations.

Multiplication of matrix 3x3 and multiplying matrices 2x2 are key topics covered. The document explains how to multiply matrices and provides a matrix multiplication step by step calculator approach. It also touches on the matrix multiplication algorithm and shows how to solve matrices step-by-step.

The content includes information on determinants, inverses, minors, and cofactors, which are essential concepts in determinants and inverses in further core one. It demonstrates how to solve matrix equations with minors and cofactors and explains the process of finding minors and cofactors of a 3x3 matrix.

Key points include:

  • Different types of matrices (zero, identity, square)
  • Matrix addition, subtraction, and multiplication
  • Determinants and their properties
  • Inverse matrices and their applications
  • Solving systems of equations using matrices

The document provides practical examples, including a problem-solving scenario involving a mole-rat colony, to illustrate the application of matrix methods in real-world situations.

9/16/2023

232

( 6 4 - 23)
zero matrix - all elements are o
identity matrix diagonot left are all ones.
adding / subtracting.
(63) (39) · (42)
03
(²6)
2x2

Problem-Solving with Matrices

This page applies the matrix concepts learned to solve a real-world problem involving a mole-rat colony. It demonstrates how to set up and solve a system of equations using matrix methods.

Example: A colony of 1000 mole-rats consists of adult males, adult females, and youngsters. After one year, the population changes as follows:

  • Adult males increase by 2%
  • Adult females increase by 3%
  • Youngsters decrease by 4%
  • Total population decreases by 20

The problem is approached by assigning variables to unknowns and translating the given information into a system of equations:

  1. M + F + Y = 1000 (initial total population)
  2. F - M = 100 (100 more females than males initially)
  3. 1.02M + 1.03F + 0.96Y = 980 (population after changes)

These equations are then converted into a matrix equation:

[1 1 1; -1 1 0; 1.02 1.03 0.96] * [M; F; Y] = [1000; 100; 980]

Highlight: The solution is found by multiplying both sides of the equation by the inverse of the coefficient matrix.

The final solution shows that the original colony consisted of:

  • 100 adult males
  • 200 adult females
  • 700 youngsters

This example illustrates the practical application of matrix multiplication step by step calculator methods and demonstrates how to solve matrix equations with minors and cofactors in a real-world context.

( 6 4 - 23)
zero matrix - all elements are o
identity matrix diagonot left are all ones.
adding / subtracting.
(63) (39) · (42)
03
(²6)
2x2

View

Matrix Inverses and 3x3 Determinants

This page delves deeper into matrix inverses and extends the concept of determinants to 3x3 matrices. It begins with a proof demonstrating that for non-singular matrices P and Q, (PQ)⁻¹ = Q⁻¹P⁻¹.

Highlight: The proof uses the properties of matrix multiplication and the definition of inverse matrices to show the relationship between the inverse of a product and the product of inverses.

The page then introduces the calculation of determinants for 3x3 matrices using the following formula:

Definition: For a 3x3 matrix M = [a b c; d e f; g h i], det(M) = a(ei-fh) - b(di-fg) + c(dh-eg)

The process of finding the inverse of a 3x3 matrix is explained, involving the following steps:

  1. Calculate the determinant
  2. Find the matrix of minors
  3. Convert to the matrix of cofactors
  4. Transpose the cofactor matrix
  5. Divide by the determinant

Example: The inverse of a 3x3 matrix A is given by A⁻¹ = (1/det(A)) * CT, where C is the cofactor matrix and CT is its transpose.

The page also introduces the concept of using matrix inverses to solve systems of equations, particularly for 3x3 systems.

Vocabulary: Cofactors are the signed minors of a matrix, used in calculating the inverse.

( 6 4 - 23)
zero matrix - all elements are o
identity matrix diagonot left are all ones.
adding / subtracting.
(63) (39) · (42)
03
(²6)
2x2

View

Matrix Operations and Applications

This page introduces fundamental concepts of matrix operations and their properties. It covers various types of matrices and basic operations such as addition, subtraction, and multiplication.

Definition: A zero matrix is a matrix where all elements are zero.

Definition: An identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere.

The page explains how to multiply matrices, emphasizing that matrix multiplication is only possible when the number of columns in the first matrix equals the number of rows in the second matrix.

Highlight: Matrix multiplication is not commutative, meaning AB ≠ BA in general.

Multiplying matrices 2x2 is demonstrated with examples, and the concept of determinants is introduced for 2x2 matrices.

Example: For a 2x2 matrix, the determinant is calculated as det(M) = ad - bc, where M = [a b; c d].

The page also touches on the concepts of singular and non-singular matrices, which are important for understanding matrix invertibility.

Vocabulary: A matrix is singular if its determinant is zero, and non-singular if its determinant is non-zero.

Finally, the page introduces the concept of matrix inverses, noting that for non-singular matrices A and B, (AB)⁻¹ = B⁻¹A⁻¹.

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

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Knowunity is the # 1 ranked education app in five European countries

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Average App Rating

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Students use Knowunity

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In Education App Charts in 12 Countries

950 K+

Students uploaded study notes

Still not sure? Look at what your fellow peers are saying...

iOS User

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

Stefan S, iOS User

The application is very simple and well designed. So far I have found what I was looking for :D

SuSSan, iOS User

Love this App ❤️, I use it basically all the time whenever I'm studying

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Knowunity is the # 1 ranked education app in five European countries

4.9+

Average App Rating

13 M

Students use Knowunity

#1

In Education App Charts in 12 Countries

950 K+

Students uploaded study notes

Still not sure? Look at what your fellow peers are saying...

iOS User

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

Stefan S, iOS User

The application is very simple and well designed. So far I have found what I was looking for :D

SuSSan, iOS User

Love this App ❤️, I use it basically all the time whenever I'm studying

How to Multiply Matrices and Solve Equations: Step by Step for Kids

user profile picture

Ahmed Nour ✓™

@ahmednour

·

412 Followers

Follow

This document covers matrix operations, determinants, and inverse matrices, with a focus on solving systems of equations using matrix methods. It provides step-by-step explanations and examples for various matrix calculations.

Multiplication of matrix 3x3 and multiplying matrices 2x2 are key topics covered. The document explains how to multiply matrices and provides a matrix multiplication step by step calculator approach. It also touches on the matrix multiplication algorithm and shows how to solve matrices step-by-step.

The content includes information on determinants, inverses, minors, and cofactors, which are essential concepts in determinants and inverses in further core one. It demonstrates how to solve matrix equations with minors and cofactors and explains the process of finding minors and cofactors of a 3x3 matrix.

Key points include:

  • Different types of matrices (zero, identity, square)
  • Matrix addition, subtraction, and multiplication
  • Determinants and their properties
  • Inverse matrices and their applications
  • Solving systems of equations using matrices

The document provides practical examples, including a problem-solving scenario involving a mole-rat colony, to illustrate the application of matrix methods in real-world situations.

9/16/2023

232

 

11th/12th

 

Math (ACT®)

9

( 6 4 - 23)
zero matrix - all elements are o
identity matrix diagonot left are all ones.
adding / subtracting.
(63) (39) · (42)
03
(²6)
2x2

Sign up to see the content. It's free!

Access to all documents

Join milions of students

Improve your grades

By signing up you accept Terms of Service and Privacy Policy

Problem-Solving with Matrices

This page applies the matrix concepts learned to solve a real-world problem involving a mole-rat colony. It demonstrates how to set up and solve a system of equations using matrix methods.

Example: A colony of 1000 mole-rats consists of adult males, adult females, and youngsters. After one year, the population changes as follows:

  • Adult males increase by 2%
  • Adult females increase by 3%
  • Youngsters decrease by 4%
  • Total population decreases by 20

The problem is approached by assigning variables to unknowns and translating the given information into a system of equations:

  1. M + F + Y = 1000 (initial total population)
  2. F - M = 100 (100 more females than males initially)
  3. 1.02M + 1.03F + 0.96Y = 980 (population after changes)

These equations are then converted into a matrix equation:

[1 1 1; -1 1 0; 1.02 1.03 0.96] * [M; F; Y] = [1000; 100; 980]

Highlight: The solution is found by multiplying both sides of the equation by the inverse of the coefficient matrix.

The final solution shows that the original colony consisted of:

  • 100 adult males
  • 200 adult females
  • 700 youngsters

This example illustrates the practical application of matrix multiplication step by step calculator methods and demonstrates how to solve matrix equations with minors and cofactors in a real-world context.

( 6 4 - 23)
zero matrix - all elements are o
identity matrix diagonot left are all ones.
adding / subtracting.
(63) (39) · (42)
03
(²6)
2x2

Sign up to see the content. It's free!

Access to all documents

Join milions of students

Improve your grades

By signing up you accept Terms of Service and Privacy Policy

Matrix Inverses and 3x3 Determinants

This page delves deeper into matrix inverses and extends the concept of determinants to 3x3 matrices. It begins with a proof demonstrating that for non-singular matrices P and Q, (PQ)⁻¹ = Q⁻¹P⁻¹.

Highlight: The proof uses the properties of matrix multiplication and the definition of inverse matrices to show the relationship between the inverse of a product and the product of inverses.

The page then introduces the calculation of determinants for 3x3 matrices using the following formula:

Definition: For a 3x3 matrix M = [a b c; d e f; g h i], det(M) = a(ei-fh) - b(di-fg) + c(dh-eg)

The process of finding the inverse of a 3x3 matrix is explained, involving the following steps:

  1. Calculate the determinant
  2. Find the matrix of minors
  3. Convert to the matrix of cofactors
  4. Transpose the cofactor matrix
  5. Divide by the determinant

Example: The inverse of a 3x3 matrix A is given by A⁻¹ = (1/det(A)) * CT, where C is the cofactor matrix and CT is its transpose.

The page also introduces the concept of using matrix inverses to solve systems of equations, particularly for 3x3 systems.

Vocabulary: Cofactors are the signed minors of a matrix, used in calculating the inverse.

( 6 4 - 23)
zero matrix - all elements are o
identity matrix diagonot left are all ones.
adding / subtracting.
(63) (39) · (42)
03
(²6)
2x2

Sign up to see the content. It's free!

Access to all documents

Join milions of students

Improve your grades

By signing up you accept Terms of Service and Privacy Policy

Matrix Operations and Applications

This page introduces fundamental concepts of matrix operations and their properties. It covers various types of matrices and basic operations such as addition, subtraction, and multiplication.

Definition: A zero matrix is a matrix where all elements are zero.

Definition: An identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere.

The page explains how to multiply matrices, emphasizing that matrix multiplication is only possible when the number of columns in the first matrix equals the number of rows in the second matrix.

Highlight: Matrix multiplication is not commutative, meaning AB ≠ BA in general.

Multiplying matrices 2x2 is demonstrated with examples, and the concept of determinants is introduced for 2x2 matrices.

Example: For a 2x2 matrix, the determinant is calculated as det(M) = ad - bc, where M = [a b; c d].

The page also touches on the concepts of singular and non-singular matrices, which are important for understanding matrix invertibility.

Vocabulary: A matrix is singular if its determinant is zero, and non-singular if its determinant is non-zero.

Finally, the page introduces the concept of matrix inverses, noting that for non-singular matrices A and B, (AB)⁻¹ = B⁻¹A⁻¹.

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Knowunity is the # 1 ranked education app in five European countries

4.9+

Average App Rating

13 M

Students use Knowunity

#1

In Education App Charts in 12 Countries

950 K+

Students uploaded study notes

Still not sure? Look at what your fellow peers are saying...

iOS User

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

Stefan S, iOS User

The application is very simple and well designed. So far I have found what I was looking for :D

SuSSan, iOS User

Love this App ❤️, I use it basically all the time whenever I'm studying