This document covers matrix operations, determinants, and inverse matrices, with... Show more
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Feb 20, 2026
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How to Multiply Matrices and Solve Equations: Step by Step for Kids
Ahmed Nour ✓™
@ahmednour
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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, (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 - b + c
The process of finding the inverse of a 3x3 matrix is explained, involving the following steps:
- Calculate the determinant
- Find the matrix of minors
- Convert to the matrix of cofactors
- Transpose the cofactor matrix
- Divide by the determinant
Example: The inverse of a 3x3 matrix A is given by A⁻¹ = * 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.
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:
- M + F + Y = 1000 (initial total population)
- F - M = 100 (100 more females than males initially)
- 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.
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.
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You can download the app in the Google Play Store and in the Apple App Store.
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App Store
4.7/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
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
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
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
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
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
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
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
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
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
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
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
iOS user
How to Multiply Matrices and Solve Equations: Step by Step for Kids
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
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 - b + c
The process of finding the inverse of a 3x3 matrix is explained, involving the following steps:
- Calculate the determinant
- Find the matrix of minors
- Convert to the matrix of cofactors
- Transpose the cofactor matrix
- Divide by the determinant
Example: The inverse of a 3x3 matrix A is given by A⁻¹ = * 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.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
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:
- M + F + Y = 1000 (initial total population)
- F - M = 100 (100 more females than males initially)
- 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.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
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.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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Students love us — and so will you.
4.6/5
App Store
4.7/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
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
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
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
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
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
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
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
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
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
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
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
iOS user