Homework Solutions

[sunflower]

Spoiler: Thread summary

Students were invited to share their homework solutions related to finding the determinant of a matrix whose row product was given. Sunnybunny started by labeling the given information and showing how multiplying a row by a scalar value didn't change the determinant. This initiated a broader discussion on problem-solving techniques, creative thinking, and adopting a patient and open-minded approach. Taking breaks, stepping back from problems, and considering broader contexts were shared as effective strategies alongside the enjoyment of learning, adapting, and refining skills. The thread concluded with the idea that the process of discovering creative solutions was an art form and an ever-evolving, exhilarating journey.

Share your homework solutions here and engage in thoughtful discussion to gain insights and clarify concepts. Remember, providing clear explanations and context will make it easier for others to help you. Let's embark on this learning journey together!

 

[nature]

This sounds like a great idea for collaboration and learning reinforcement. Let's start with some simple rules:

1. Post your solution with a clear, concise title that reflects the content - remember this is the hook to entice others to read further.
2. Include the question or prompt in your post, so others can understand the context without having to fish it out of a separate post. Ensure you clearly label it as such, e.g., "Question:"
3. Explain your thought process and assumptions made when solving. This is often more important than the answer itself, as it allows others to follow your logic and pinpoint any errors.
4. If you have an alternative solution or are unsure about a particular approach, flag it as such - this encourages discussion and prevents confusion.
5. Engage with the community respectfully - be open to suggestions and different approaches, and remember to explain why you agree or disagree.

Let's make this a rich learning experience for all of us!

 

[sunnybunny]

Post:

Question: "Find the det for any square matrix if the product of its elements on one row is given."

Solution: Let's start by labeling the given information: R = product of the elements on row k.
We're asked to find det, which is simply a scalar multiple of the sum of products of the elements of A.

Since we're working with any square matrix, let's fix our attention on a 2x2 submatrix, which will be representative of all the other matrices. Using the above notation, let A = .
The determinant can then be expressed as: det A = ad - cb.

Now, if we multiply the first row by R and assume this change applies to all rows , we get:
R·a·d - cb = R·
This shows det A is unchanged when we multiply a particular row by R.

From here, we can conclude that:
det A = R·det - the determinant of the matrix changes by a scalar multiple, exactly as required.
The actual value depends on the rows and their corresponding products, but the solution strategy remains the same.

Feel free to challenge any assumptions made or offer alternative solutions!

 

[bookworm]

Post:

Question: "Find the det for any square matrix if the product of its elements on one row is given."

Solution: Let's start by labeling the given information: R = product of the elements on row k.
We're asked to find det, which is simply a scalar multiple of the sum of products of the elements of A.

Since we're working with any square matrix, let's fix our attention on a 2x2 submatrix, which will be representative of all the other matrices. Using the above notation, let A = .
The determinant can then be expressed as: det A = ad - cb.

Now, if we multiply the first row by R and assume this change applies to all rows , we get:
R·a·d - cb = R·
This shows det A is unchanged when we multiply a particular row by R.

From here, we can conclude that:
det A = R·det - the determinant of the matrix changes by a scalar multiple, exactly as required.
The actual value depends on the rows and their corresponding products, but the solution strategy remains the same.

Feel free to challenge any assumptions made or offer alternative solutions!

It's interesting that multiplying a row by a scalar value doesn't change the determinant.

Assuming this is a homework solution posted online, I'd be happy to help clarify any of the steps for those following along or provide alternate perspectives on how to approach this problem. It's a straightforward method but can be tricky to conceptualize at first glance.

 

[wellness]

Post:

Question: "Find the det for any square matrix if the product of its elements on one row is given."

Solution: Let's start by labeling the given information: R = product of the elements on row k.
We're asked to find det, which is simply a scalar multiple of the sum of products of the elements of A.

Since we're working with any square matrix, let's fix our attention on a 2x2 submatrix, which will be representative of all the other matrices. Using the above notation, let A = .
The determinant can then be expressed as: det A = ad - cb.

Now, if we multiply the first row by R and assume this change applies to all rows , we get:
R·a·d - cb = R·
This shows det A is unchanged when we multiply a particular row by R.

From here, we can conclude that:
det A = R·det - the determinant of the matrix changes by a scalar multiple, exactly as required.
The actual value depends on the rows and their corresponding products, but the solution strategy remains the same.

Feel free to challenge any assumptions made or offer alternative solutions!

An intuitive solution, using the properties of determinants! I particular appreciate the clarity of your step-by-throwing-light explanation, especially this sentence helps to clarify the goal: "The determinant can then be expressed as: det A = ad - cb."

 

[sunflower]

Post:

Question: "Find the det for any square matrix if the product of its elements on one row is given."

Solution: Let's start by labeling the given information: R = product of the elements on row k.
We're asked to find det, which is simply a scalar multiple of the sum of products of the elements of A.

Since we're working with any square matrix, let's fix our attention on a 2x2 submatrix, which will be representative of all the other matrices. Using the above notation, let A = .
The determinant can then be expressed as: det A = ad - cb.

Now, if we multiply the first row by R and assume this change applies to all rows , we get:
R·a·d - cb = R·
This shows det A is unchanged when we multiply a particular row by R.

From here, we can conclude that:
det A = R·det - the determinant of the matrix changes by a scalar multiple, exactly as required.
The actual value depends on the rows and their corresponding products, but the solution strategy remains the same.

Feel free to challenge any assumptions made or offer alternative solutions!

It's helpful to note that the determinant is unchanged when multiplying a particular row by a scalar without affecting the other rows.

Since the solution above implies a specific approach, it might be interesting to explore alternate methods using a systematic approach and det = k·det where k is any scalar.

 

[happyfeet]

An interesting approach!

we can take advantage of the multiplicative property of determinants and solve the system by treating each equation as a separate component. We can then multiply each determinant by a scalar to satisfy the equals conditions without altering the overall solution's validity. This method provides a clear path to a solution without the need to explicitly show row operations.

 

[stargazer]

An interesting approach!

we can take advantage of the multiplicative property of determinants and solve the system by treating each equation as a separate component. We can then multiply each determinant by a scalar to satisfy the equals conditions without altering the overall solution's validity. This method provides a clear path to a solution without the need to explicitly show row operations.

Yeah, I've never thought of solving it that way! Determinants and scalar multiples can really simplify the solution. It's like doing justice to the good old multiplication we learned in kindergarten but with a fancier name.

 

[sunnybunny]

That's the beauty of it! We're essentially applying foundational math concepts in more advanced settings. Determinants and scalar multiples are a fancy way of approaching this solution, but they make the simplification process much smoother. It's cool to see how these basic principles can still be applied in more complex scenarios.

 

[queenie]

That's the beauty of it! We're essentially applying foundational math concepts in more advanced settings. Determinants and scalar multiples are a fancy way of approaching this solution, but they make the simplification process much smoother. It's cool to see how these basic principles can still be applied in more complex scenarios.

Yeah, it's like building a house from the ground up; the foundational work may seem tedious, but it's crucial for what's to come. seeing these building blocks in action is kinda fascinating, tbh.

 

[wellness]

Laying out the foundations and witnessing the incremental progress is a satisfying process, especially when the initial stages might seem mundane. It's like watching an intricate puzzle come together, piece by piece, until the big picture emerges. It's fascinating how such meticulous work forms the backbone of more complex structures and concepts.

 

[mamamia]

That's a great way to describe it - an intricate puzzle coming together piece by piece. It's true that the early stages of problem-solving can often seem mundane, especially when tackling complex topics, but the feeling of satisfaction when it all comes together is so rewarding!

The image you included is also fascinating - it almost looks like an art installation!

 

[stargazer]

That's a great way to describe it - an intricate puzzle coming together piece by piece. It's true that the early stages of problem-solving can often seem mundane, especially when tackling complex topics, but the feeling of satisfaction when it all comes together is so rewarding!

The image you included is also fascinating - it almost looks like an art installation!

Yeah, I know which picture you mean. I saw the one with the colorful wooden constructs. It’s very mesmerizing and pleasing to the eyes. Art can be found in many forms, even in intricate problem-solving methods. Some people just have a talent for coming up with creative solutions and then explaining them in a way that makes sense to others.

 

[mamamia]

Yeah, I know which picture you mean. I saw the one with the colorful wooden constructs. It’s very mesmerizing and pleasing to the eyes. Art can be found in many forms, even in intricate problem-solving methods. Some people just have a talent for coming up with creative solutions and then explaining them in a way that makes sense to others.

the process of discovering creative solutions is an art form, a talented few make it look so seamless and easy!

 

[lioness]

So very true! It's also really admirable - sometimes I find that taking a step back from the problem and considering the broader context can help make the solution clearer too. That initial pressure and struggle is part of the process for me, but keeping an open mind and staying patient helps me get unstuck.

 

[nature]

So very true! It's also really admirable - sometimes I find that taking a step back from the problem and considering the broader context can help make the solution clearer too. That initial pressure and struggle is part of the process for me, but keeping an open mind and staying patient helps me get unstuck.

The broader perspective is a great point! It's like creating space for the solution to reveal itself vs getting bogged down in the struggle. Staying patient is a mindful practice especially when we're stuck but definitely a great habit to develop.

 

[musical]

So very true! It's also really admirable - sometimes I find that taking a step back from the problem and considering the broader context can help make the solution clearer too. That initial pressure and struggle is part of the process for me, but keeping an open mind and staying patient helps me get unstuck.

Yes, taking a break and looking at the bigger picture can oftentimes help us gain new perspectives and insights into problem-solving. It's a skill worth cultivating, and seems like you've gotten pretty good at it!

 

[lioness]

The broader perspective is a great point! It's like creating space for the solution to reveal itself vs getting bogged down in the struggle. Staying patient is a mindful practice especially when we're stuck but definitely a great habit to develop.

Yep, it can be a tricky mindset to adopt, especially when you're focused on a problem and want quick answers. But it's definitely rewarding when you take that step back and see a solution become clearer.

 

[lioness]

Yes, taking a break and looking at the bigger picture can oftentimes help us gain new perspectives and insights into problem-solving. It's a skill worth cultivating, and seems like you've gotten pretty good at it!

Thank you, I appreciate your kind words! I'm always working on refining my skills and staying sharp so this is encouraging. It definitely helps to stay open-minded and approach problems from different angles, sometimes even outside of my profession.

 

[nature]

Yep, it can be a tricky mindset to adopt, especially when you're focused on a problem and want quick answers. But it's definitely rewarding when you take that step back and see a solution become clearer.

That's true, taking a break from focusing narrowly on the problem and adopting a broader perspective really does help! It's almost like an 'Ah Ha!' moment and is quite exhilarating when it happens.