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

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Ahmed Nour ✓™

9/16/2023

Math (ACT®)

Matrices

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Math (ACT®)

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Preparation and Review

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Matrices and Vectors

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

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

287

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

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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, PQPQ⁻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 = abc;def;ghia b c; d e f; g h i, detMM = aei−fhei-fh - bdi−fgdi-fg + cdh−egdh-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(A1/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

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 initialtotalpopulationinitial total population
  2. F - M = 100 100morefemalesthanmalesinitially100 more females than males initially
  3. 1.02M + 1.03F + 0.96Y = 980 populationafterchangespopulation after changes

These equations are then converted into a matrix equation:

111;−110;1.021.030.961 1 1; -1 1 0; 1.02 1.03 0.96 * M;F;YM; F; Y = 1000;100;9801000; 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.

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Operations, determinants, inverses, properties, simultaneous equations, transformations, invariance

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Subjects

Math (ACT®)

Preparation and Review

Matrices and Vectors

 

Math (ACT®)

•

287

•

Sep 16, 2023

•

3 pages

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

user profile picture

Ahmed Nour ✓™

@ahmednour

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 2x2are key... Show more

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

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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, PQPQ⁻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 = abc;def;ghia b c; d e f; g h i, detMM = aei−fhei-fh - bdi−fgdi-fg + cdh−egdh-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(A1/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 contentIt's free!

Access to all documents

Improve your grades

Join milions of students

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 initialtotalpopulationinitial total population
  2. F - M = 100 100morefemalesthanmalesinitially100 more females than males initially
  3. 1.02M + 1.03F + 0.96Y = 980 populationafterchangespopulation after changes

These equations are then converted into a matrix equation:

111;−110;1.021.030.961 1 1; -1 1 0; 1.02 1.03 0.96 * M;F;YM; F; Y = 1000;100;9801000; 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

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 detMM = ad - bc, where M = ab;cda 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, ABAB⁻¹ = B⁻¹A⁻¹.

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This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

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Anna

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I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

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David K

iOS user

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Sudenaz Ocak

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Greenlight Bonnie

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Marco B

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THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

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This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

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