Solving a determinant using another determinant

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In summary, the conversation discusses a question about finding the determinant of a matrix and the speaker is unsure of how to approach it as there is no clear solution. They provide two examples that result in different determinants for the given matrices.
  • #1
mrhobbes
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Homework Statement



Hi everyone. I'm just going over some questions for a midterm and came upon one that I don't seem to understand. The question is follows

Assuming a determinant
|a b c |
|d e f | = 4
|g h i |

Find the determinant if
|3a 2b 4c|
|d e f | = ?
|g h i |

The Attempt at a Solution



I'm honestly stuck on this question and have no idea where to begin.
 
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  • #2
I have no idea what is wanted here. There is no manipulation of rows and columns that will reduce the second matrix to the first and there is no single numerical answer.

For example, a= 4, e= i= 1, all other entries 0, gives 4 for the first determinant and 12 for the second.

But b= -4, d= 1, i= 1, all other entries 0, also gives 4 for the first determinant but 8 for the second.
 

1. How do I solve a determinant using another determinant?

To solve a determinant using another determinant, you will need to use the properties of determinants, such as row operations and expansion by minors. The process involves manipulating the given determinant into a form where you can use the properties to simplify it. This may involve using the properties multiple times until the determinant is reduced to a scalar value.

2. Why would I need to solve a determinant using another determinant?

Solving a determinant using another determinant can be useful in various mathematical applications, such as solving systems of linear equations and finding the inverse of a matrix. It can also be used to simplify complex expressions involving determinants.

3. What are some common mistakes when solving a determinant using another determinant?

Some common mistakes when solving a determinant using another determinant include forgetting to apply the properties of determinants correctly, making errors in the calculations, and not keeping track of the operations performed. It is important to double-check your work and be careful with the calculations to avoid these mistakes.

4. Is there a specific order in which the properties of determinants should be applied?

No, there is no specific order in which the properties of determinants should be applied. It depends on the given determinant and the preferred method of solving it. However, it is important to keep track of the operations performed to avoid making mistakes and to ensure a correct solution.

5. Can I use any other methods to solve a determinant besides using another determinant?

Yes, there are other methods for solving determinants, such as Cramer's rule, Gaussian elimination, and using the LU decomposition. However, using another determinant is a common and efficient method for solving determinants, especially for larger matrices.

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